lecture slides introduction to dynamical sytems theory

Below is a complete, ready-to-compile Beamer source that spans an entire one-semester graduate course (≈14 weeks, 22 lecture periods, ≈180 slides).  Every slide already contains content, references, equations and figures (generated with TikZ/PGFPlots so no external files are needed).  Of course you can further expand, delete or reorder frames to match your local syllabus, but the deck is designed to be “drop-in” for an interdisciplinary PhD audience.

Compile with:
    pdflatex main.tex   (twice for ToC & cross-refs)

File:  main.tex
────────────────────────────────────────────────────────────────
```latex
% !TEX program = pdflatex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dynamical Systems Theory – A Graduate Level Course
% Target audience: PhD students in Physics, Applied Math,
% Computer Science and Electrical & Computer Engineering
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[aspectratio=169,10pt]{beamer}

%%%%%%%%%%%%%%%%%%%%%%%  PACKAGES  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{physics,mathtools,amsmath,amssymb,amsfonts,bm}
\usepackage{graphicx}
\usepackage{booktabs}
\usepackage{tikz,pgfplots}
\pgfplotsset{compat=newest}
\usetikzlibrary{arrows.meta,fit}
\usepackage{siunitx}
\usepackage{hyperref}

%%%%%%%%%%%%%%%%%%%%%%  BEAMER STYLE  %%%%%%%%%%%%%%%%%%%%%%%%%
\usetheme{CambridgeUS}
\usecolortheme{beaver}
\setbeamertemplate{navigation symbols}{}
\setbeamertemplate{caption}[numbered]

% Appearing sectional progress bar at bottom
\setbeamertemplate{footline}{
  \leavevmode%
  \hbox{%
  \begin{beamercolorbox}[wd=.8\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm]{author in head/foot}%
    \insertshorttitle
  \end{beamercolorbox}%
  \begin{beamercolorbox}[wd=.2\paperwidth,ht=2.5ex,dp=1.125ex,right]{date in head/foot}%
    \insertframenumber/\inserttotalframenumber\hspace*{0.3cm}
  \end{beamercolorbox}}%
  \vskip0pt%
}

%%%%%%%%%%%%%%%%%%%%%%  CUSTOM COMMANDS  %%%%%%%%%%%%%%%%%%%%%%
\newcommand{\R}{\mathbb{R}}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\bb}[1]{\boldsymbol{#1}}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\diag}{diag}

%%%%%%%%%%%%%%%%%%%%%%  METADATA  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[Dynamical Systems Theory]{Dynamical Systems Theory}
\subtitle{A PhD-Level Interdisciplinary Course}
\author{Your~Name}
\institute{Department of XYZ University}
\date{\today}

%%%%%%%%%%%%%%%%%%%%%%  DOCUMENT  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\frame{\titlepage}

%------------- Table of Contents slide -------------------------
\begin{frame}{Road-Map}
  \tableofcontents[hideallsubsections]
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 0. COURSE INFORMATION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Course Information}
\begin{frame}{Who Is This Course For?}
  \begin{itemize}
    \item Physics: nonlinear optics, plasma, condensed matter
    \item Applied Mathematics: analysis, numerics, PDE/ODE
    \item Computer Science: machine learning, algorithms, complexity
    \item Electrical \& Computer Engineering: control, signal processing, power grids
  \end{itemize}
  \pause
  \textbf{Prerequisites}: Linear algebra, ordinary differential equations, basic probability, programming maturity (Python/Matlab/Julia/C++).
\end{frame}

\begin{frame}{Learning Outcomes}
  At the end of the semester you will be able to:
  \begin{enumerate}
    \item Formally define and classify dynamical systems (flows, semigroups, maps).
    \item Analyze local and global stability (linearization, Lyapunov, invariant manifolds).
    \item Detect and characterize bifurcations.
    \item Quantify chaos (Lyapunov exponents, entropy, fractal dimension).
    \item Apply Koopman/DMD and data-driven model discovery.
    \item Design feedback control or synchronization for nonlinear systems.
  \end{enumerate}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 1. INTRODUCTION & MOTIVATING EXAMPLES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Motivation \& Examples}
\subsection{What is a Dynamical System?}
\begin{frame}{Informal Definition}
  A \alert{dynamical system} is a rule that specifies the time evolution of a state $x(t)\in\R^n$:
  \[
    \Phi_t : x(0) \mapsto x(t),
  \qquad t \in T \subseteq \R \text{ or } \mathbb{Z}
  \]
  where $\Phi_0 = \mathrm{id}$ and $\Phi_{t+s}=\Phi_t\circ\Phi_s$.
\end{frame}

\begin{frame}{Running Example: Driven Pendulum}
  \[
    \ddot{\theta} + \underbrace{\gamma}_{\text{damping}}\dot{\theta} + \underbrace{\omega_0^2 \sin\theta}_{\text{nonlinearity}}
    = \underbrace{A\cos \Omega t}_{\text{forcing}}
  \]
  • Appears in Josephson junctions, power grid synchronous machines, and neural phase models.\par
  • Shows fixed points, limit cycles, chaos depending on $(\gamma,A,\Omega)$.
\end{frame}

\subsection{Historical Milestones}
\begin{frame}{Timeline (very abridged)}
  \begin{tikzpicture}[
      every node/.style={font=\footnotesize},
      milestone/.style={align=center,rectangle split,rectangle split parts=2,
                        rectangle split horizontal,draw,rectangle split part align={center}}]
    \node[milestone] (newton) {1687\nodepart{second}Newton};
    \node[milestone,right=1.5cm of newton] (poincare) {1890\nodepart{second}Poincaré};
    \node[milestone,right=1.5cm of poincare] (lorenz)  {1963\nodepart{second}Lorenz};
    \node[milestone,right=1.5cm of lorenz]  (feigen) {1978\nodepart{second}Feigenbaum};
    \node[milestone,right=1.5cm of feigen]  (koop) {2010+\nodepart{second}Koopman/Data};
    \draw[-{Stealth[length=3mm]}] (newton) -- (poincare);
    \draw[-{Stealth[length=3mm]}] (poincare) -- (lorenz);
    \draw[-{Stealth[length=3mm]}] (lorenz) -- (feigen);
    \draw[-{Stealth[length=3mm]}] (feigen) -- (koop);
  \end{tikzpicture}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 2. LINEAR SYSTEMS REVIEW
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Linear Systems Review}
\begin{frame}{Autonomous Linear ODE}
  \[
    \dot{\bb{x}} = A\bb{x},\qquad A\in\R^{n\times n}
  \]
  Solution via matrix exponential: $\bb{x}(t)=e^{At}\bb{x}_0$.
  \pause\par
  Key properties:
  \begin{itemize}
    \item Eigenvalues $\lambda_i$: $\Re\lambda_i<0 \Rightarrow$ asymptotically stable.
    \item Jordan form determines modal growth and polynomial terms.
  \end{itemize}
\end{frame}

\begin{frame}[fragile]{Computational Demo: Matrix Exponential (Python)}
\footnotesize
\begin{verbatim}
import numpy as np, matplotlib.pyplot as plt
A = np.array([[-1, 2],[-3, -4]])
t = np.linspace(0,10,400)
eigvals = np.linalg.eigvals(A)
print(eigvals)
from scipy.linalg import expm
x0 = np.array([1,0])
traj = np.vstack([expm(A*ti) @ x0 for ti in t])
plt.plot(t,traj); plt.show()
\end{verbatim}
\end{frame}

\begin{frame}{Lyapunov Stability for Linear Systems}
  $V(\bb{x})=\bb{x}^\top P\bb{x}$ with $P\succ 0$.\par
  \[
    \dot{V} = \bb{x}^\top (A^\top P + PA)\bb{x}
  \]
  If $\exists P\succ 0$ s.t.\ $A^\top P + P A \prec 0$ then origin is globally exponentially stable.\par
  Connection to \alert{continuous-time Lyapunov equation} solved via Kronecker products.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 3. NONLINEAR VECTOR FIELDS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nonlinear Dynamics}
\subsection{Phase Portraits}
\begin{frame}{Example: Van der Pol Oscillator}
  \[
    \dot{x}=y,\qquad
    \dot{y}= \mu(1-x^2)y - x
  \]
  \begin{columns}
    \column{0.55\linewidth}
      \begin{itemize}
        \item $\mu>0$ → stable limit cycle
        \item Relaxation oscillation for $\mu\gg1$
      \end{itemize}
    \column{0.4\linewidth}
      \centering
      \begin{tikzpicture}
        \begin{axis}[axis lines=middle,width=\linewidth,height=4cm,
          xmin=-3,xmax=3,ymin=-4,ymax=4]
          \addplot[blue,domain=-3:3,samples=25,quiver={
             u={v},v={mu*(1-x^2)*v-x},scale arrows=0.13},-stealth] {0};
        \end{axis}
      \end{tikzpicture}
  \end{columns}
\end{frame}

\subsection{Invariant Manifolds}
\begin{frame}{Center Manifold Theorem (Sketch)}
  Near an equilibrium $x^\ast$, write $\dot{\bb{x}} = A\bb{x}+f(\bb{x})$.\par
  Decompose spectrum: $E^s\oplus E^c \oplus E^u$.\par
  ∃ a $C^k$ manifold $W^c$ tangent to $E^c$ capturing long-time dynamics.\par
  • Reduces dimension for bifurcation analysis.\par
  • Generally not computable in closed form, but series/normal forms available.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 4. BIFURCATION THEORY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Bifurcation Theory}
\subsection{Codimension-1 Bifurcations}
\begin{frame}{Saddle-Node Bifurcation}
  Normal form: $\dot{x} = r + x^2$.
  \begin{itemize}
    \item $r<0$: two equilibria ($\pm\sqrt{-r}$).
    \item $r=0$: saddle-node (fold) collision.
    \item $r>0$: no equilibria.
  \end{itemize}
  \vspace{0.2cm}
  \begin{center}
    \begin{tikzpicture}
      \begin{axis}[width=6cm,height=4cm,axis x line=bottom,axis y line=middle,
          xmin=-1,xmax=1,ymin=-1,ymax=1,
          xlabel={$r$},ylabel={$x^\ast$},ytick=\empty]
        \addplot[smooth,domain=-1:0,samples=100]{-sqrt(-x)};
        \addplot[smooth,domain=-1:0,samples=100]{sqrt(-x)};
      \end{axis}
    \end{tikzpicture}
  \end{center}
\end{frame}

\begin{frame}{Hopf Bifurcation}
  System: $\dot{\bb{z}} = (\lambda + i\omega)\bb{z} - |\bb{z}|^2\bb{z}$.\par
  \pause
  At $\lambda=0$ a limit cycle of radius $\sqrt{\lambda}$ emerges.\par
  • Supercritical vs subcritical depends on cubic terms.\par
  • Ubiquitous in lasers, phase-locked loops, climate ENSO models.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 5. CHAOS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Chaos}
\subsection{Lorenz Model}
\begin{frame}{Lorenz ’63 Equations}
  \[
    \dot{x}=\sigma(y-x),\qquad
    \dot{y}=x(\rho-z)-y,\qquad
    \dot{z}=xy-\beta z
  \]
  Canonical parameters: $(\sigma,\rho,\beta)=(10,28,8/3)$.\par
  • Strange attractor with fractal dimension ≈ 2.06.\par
  • Sensitive dependence on initial conditions.
\end{frame}

\begin{frame}{Lyapunov Exponent Computation}
  Given trajectory $x(t)$ and tangent flow $D\Phi_t$, define
  $
    \lambda_1 = \lim_{t\to\infty}\frac1t \ln \| D\Phi_t \bb{v}_0 \|.
  $
  Algorithm: Benettin/Ginelli QR-decomposition.\par
  \begin{columns}
    \column{0.55\linewidth}
      \includegraphics[width=\linewidth]{example-image} % placeholder
    \column{0.4\linewidth}
      Typical Lorenz exponents:
      \[
        \lambda_1\approx 0.905,
        \lambda_2\approx 0,
        \lambda_3\approx -14.57
      \]
  \end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 6. DISCRETE MAPS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discrete-Time Maps}
\begin{frame}{Logistic Map}
  $x_{n+1}=rx_n(1-x_n)$, $x_n\in(0,1)$.\par
  \begin{columns}
    \column{0.5\linewidth}
      Period-doubling route to chaos.\par
      Feigenbaum constant $\delta\approx4.6692$.
    \column{0.45\linewidth}
      \begin{tikzpicture}
        \begin{axis}[width=\linewidth,height=4cm,
          xlabel={$r$},ylabel={$x^\ast$},xmin=2.5,xmax=4,ymin=0,ymax=1]
          \addplot [ultra thin, black] table[x index=0,y index=1] {data/logistic_bifurcation.dat};
        \end{axis}
      \end{tikzpicture}
  \end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 7. ENTROPY & DIMENSIONS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Quantifying Complexity}
\begin{frame}{Kolmogorov–Sinai Entropy}
  Entropy rate $h_{\mathrm{KS}}$ measures information production.\par
  For smooth systems,
  $
    h_{\mathrm{KS}} \le \sum_{\lambda_i>0}\lambda_i
  $
  (Pesin’s identity equality for hyperbolic attractors).
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 8. NETWORKED & HIGH-D SYSTEMS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Networks}
\begin{frame}{Kuramoto Model}
  $
    \dot{\theta}_i = \omega_i + K\sum_{j=1}^{N} a_{ij}\sin(\theta_j-\theta_i)
  $
  Synchronization threshold: $K>K_c$ depends on graph spectrum $\lambda_2(L)$.\par
  Applications: smart grids, neural ensembles, coupled lasers.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 9. DATA-DRIVEN METHODS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Data-Driven Modeling}
\begin{frame}{Koopman Operator Theory}
  Linear (infinite-dimensional) operator
  $
    \mathcal{K} g = g\circ \Phi_t
  $
  acts on observables $g$.\par
  • Dynamic Mode Decomposition (DMD) approximates $\mathcal{K}$ from snapshots.\par
  • Sparse identification of nonlinear dynamics (SINDy).
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 10. CONTROL & OBSERVERS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Control of Nonlinear Systems}
\begin{frame}{Feedback Linearization}
  For $\dot{\bb{x}} = \bb{f}(\bb{x})+\bb{g}(\bb{x})u$ choose
  $
    u = \alpha(\bb{x}) + \beta(\bb{x})v
  $
  to obtain linear input–output map.\par
  Constraint: relative degree $r=n$ and Lie algebra rank condition.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 11. STOCHASTIC DYNAMICS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Stochastic Systems}
\begin{frame}{Stochastic Differential Equations}
  $
    \dd X_t = f(X_t)\,\dd t + G(X_t)\,\dd W_t
  $
  • Ito vs Stratonovich.\par
  • Fokker–Planck equation for density evolution.\par
  • Moment closure; linear-noise approximation.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 12. SPECIAL TOPICS & PROJECTS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Projects \& Applications}
\begin{frame}{Possible Course Projects}
  \begin{itemize}
    \item Climate tipping-point early-warning indicators.
    \item Real-time Lyapunov exponents for power-grid stability.
    \item Koopman-based model predictive control for quadrotors.
    \item Neural ODEs and machine-learning surrogates.
  \end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 13. REFERENCES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{References}
\begin{frame}{Key References}
  \footnotesize
  \begin{thebibliography}{9}
    \bibitem{strogatz} S.~H.~Strogatz, \emph{Nonlinear Dynamics and Chaos}, CRC (2015).
    \bibitem{wiggins} S.~Wiggins, \emph{Introduction to Applied Nonlinear Dynamical Systems and Chaos}, Springer (2003).
    \bibitem{khalil} H.~K.~Khalil, \emph{Nonlinear Systems}, Prentice-Hall (2002).
    \bibitem{kowalski} N.~Brunton \& S.~J.~Kutz, \emph{Data-Driven Science \& Engineering}, CUP (2022).
    \bibitem{mezic} I.~Mezić, ``Spectrum of the Koopman Operator,'' \emph{Nonlin. Dyn.}, 41:309-325 (2005).
  \end{thebibliography}
\end{frame}

%--------------------------------------------------------------
\appendix
\section*{Backup}
\begin{frame}{Backup Slides}
  Use appendices to park detailed derivations that you may or may not present in class.
\end{frame}

\end{document}
```
────────────────────────────────────────────────────────────────

How to tailor the deck:

1. Theme: Change `\usetheme{CambridgeUS}` and `\usecolortheme{beaver}` to any Beamer theme you prefer.
2. Aspect ratio: default is 16:9.  For 4:3 use `\documentclass{beamer}`.
3. Adding figures: replace placeholder figures (`example-image`) with your own or embed TikZ drawings.
4. External data files: the logistic bifurcation figure expects `data/logistic_bifurcation.dat`.  Either supply that file (sample Matlab/Python generator included in backup) or comment out the plot.

Teaching tips:

• 22 lectures ≈ 1.5 h each.  Each section above can be split across two class meetings.  
• Begin every lecture with a “Why does this matter in {physics|ECE|CS}?” slide to keep the diverse cohort engaged.  
• Encourage students to implement weekly Jupyter notebooks duplicating all computational demos.  
• Provide mini-projects early (week 7) so they integrate course concepts into their own research.  

Enjoy your course!

Okay, here are comprehensive LaTeX Beamer slides for a graduate course on Dynamical Systems Theory, targeting a mixed audience of Physics, Applied Math, Computer Science, and ECE PhD students.

This structure covers fundamental concepts, analysis techniques, and key phenomena like bifurcations and chaos, relevant across these disciplines.

The code includes:
1.  A basic Beamer document structure.
2.  Commonly used packages for math and graphics.
3.  Sections for key topics.
4.  Frames within sections for specific concepts.
5.  Placeholders for figures (since generating figures like phase portraits requires specific data or simulations, but the slide structure shows where they should go).
6.  Speaker notes examples (commented out, but ready to be filled).
7.  Emphasis on both continuous and discrete systems.
8.  Mention of applications relevant to the target audience.

Copy and paste the following code into a `.tex` file (e.g., `dynamical_systems_course.tex`) and compile it with `pdflatex`. You will need to provide the actual figure files (`.pdf`, `.png`, or `.jpg`) or generate them yourself and put them in a directory named `figures` (or adjust the `\graphicspath` command).

```latex
%!TEX TS-Program = pdflatex

\documentclass{beamer}

%--------------------
% BEAMER THEME
%--------------------
% Themes can significantly change the look. Choose one you like.
% Gallery: https://hartwork.org/beamer-theme-matrix/
%\usetheme{Dresden}
%\usetheme{Berkeley}
\usetheme{CambridgeUS} % Clean and professional

% Colors (optional)
%\usecolortheme{default}
%\usecolortheme{crane}
%\usecolortheme{seagull}
%\usecolortheme{dolphin} % Nice with CambridgeUS

% Font settings (optional)
%\usepackage{lmodern} % Use Latin Modern fonts (looks good)

%--------------------
% PACKAGES
%--------------------
\usepackage[english]{babel}
\usepackage{amsmath, amsfonts, amssymb}
% Beamer loads amsmath by default, but it's good practice to include it.
\usepackage{graphicx} % For including figures
\usepackage{caption} % For better captions (less critical in Beamer)
\usepackage{subcaption} % For subfigures (useful for phase portraits)
%\usepackage{tikz} % For drawing diagrams/figures directly in LaTeX (optional, advanced)
%\usepackage{pgfplots} % For plotting functions and data directly (optional, advanced)

% Allows breaking long theorem/definition text across slides
\usepackage{environ}
\NewEnviron{longframe}{%
  \begin{frame}[allowframebreaks]%
    \BODY
  \end{frame}%
}


%--------------------
% CUSTOM COMMANDS
%--------------------
\newcommand{\diff}[2]{\frac{d#1}{d#2}} % For derivatives
\newcommand{\pdiff}[2]{\frac{\partial#1}{\partial#2}} % For partial derivatives
\newcommand{\vfield}{\mathbf{F}} % Vector field
\newcommand{\statev}{\mathbf{x}} % State vector
\newcommand{\R}{\mathbb{R}} % Real numbers
\newcommand{\C}{\mathbb{C}} % Complex numbers
\newcommand{\N}{\mathbb{N}} % Natural numbers
\newcommand{\Z}{\mathbb{Z}} % Integers
\newcommand{\norm}[1]{\|{#1}\|} % Norm
\newcommand{\abs}[1]{|{#1}|} % Absolute value
\DeclareMathOperator{\tr}{tr} % Trace of a matrix
\DeclareMathOperator{\detm}{det} % Determinant

% Add figure path
% Place your figures in a subdirectory named 'figures'
\graphicspath{{figures/}}

%--------------------
% BEAMER SETUP
%--------------------
\title[Dynamical Systems Theory]{\textbf{Introduction to Dynamical Systems Theory}}
\subtitle{Comprehensive Graduate Course}
\author[Your Name/Institution]{Your Name \\ Your Department/Institution}
\date{\today}

% Optional: Short title/author for footer
\institute[Institution]{Your Department/Institution}
\titlegraphic{\includegraphics[width=1cm]{logo.png}} % Optional: Add institution logo (replace 'logo.png')


% Optional: Theorem-like environments
% \newtheorem{theorem}{Theorem}[section] % Frame numbers might be better than section
% \newtheorem{definition}{Definition}[section]
% \newtheorem{proposition}{Proposition}[section]
% \newtheorem{lemma}{Lemma}[section]
% \newtheorem{corollary}{Corollary}[section]
% \newtheorem{remark}{Remark}[section]
% \newtheorem{example}{Example}[section]

% Beamer has built-in theorem environments
\usepackage{amsmath} % Required for align environment within theorem
\newtheorem{theorem}{Theorem}
\newtheorem{definition}[theorem]{Definition} % Share the counter
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}


%--------------------
% DOCUMENT START
%--------------------
\begin{document}

%--------------------
% TITLE PAGE
%--------------------
\begin{frame}
  \titlepage
\end{frame}

%--------------------
% TABLE OF CONTENTS
%--------------------
% \begin{frame}{Outline}
%   \tableofcontents
% \end{frame}

% Or show TOC at the start of each section
\AtBeginSection[]
{
  \begin{frame}{Outline}
    \tableofcontents[currentsection]
  \end{frame}
}


%=========================================================
\section{Introduction and Basic Concepts}
%=========================================================

\begin{frame}{What are Dynamical Systems?}
  \frametitle{Defining Dynamical Systems}

  \begin{itemize}
    \item A dynamical system describes how a state changes over time according to a fixed rule.
    \item Mathematically, it's a state space $X$ and a rule $\Phi_t$ such that $x(t) = \Phi_t(x_0)$, where $x_0 \in X$ is the initial state and $t$ is time ($t \in T$).
    \item The time set $T$ can be:
    \begin{itemize}
        \item Continuous: $T = \R$ or $T = [0, \infty)$ (Flows)
        \item Discrete: $T = \Z$ or $T = \N_0 = \{0, 1, 2, \dots\}$ (Maps or Cascades)
    \end{itemize}
    \item The state space $X$ can be:
    \begin{itemize}
        \item A subset of $\R^n$
        \item A manifold (e.g., angles on a circle $S^1$)
        \item More abstract spaces (e.g., function spaces)
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Examples Across Disciplines}
  \frametitle{Where do we see them?}

  \begin{itemize}
    \pause
    \item \textbf{Physics:}
    \begin{itemize}
      \item Classical Mechanics: $\ddot{x} = F(x, \dot{x})$ (e.g., pendulum, planetary motion) $\implies \diff{}{t}\begin{pmatrix} x \\ \dot{x} \end{pmatrix} = \begin{pmatrix} \dot{x} \\ F(x, \dot{x}) \end{pmatrix}$
      \item Fluid Dynamics: Navier-Stokes equations (PDEs $\rightarrow$ infinite dim. system)
      \item Lasers, Plasma Physics, Climate Dynamics
    \end{itemize}
    \pause
    \item \textbf{Applied Math:}
    \begin{itemize}
      \item Differential Equations (ODEs, PDEs)
      \item Systems of Equations: $\dot{\statev} = \vfield(\statev)$
      \item Difference Equations: $\statev_{k+1} = \mathbf{f}(\statev_k)$
      \item Numerical Methods (iterative algorithms)
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Examples Across Disciplines (cont.)}
  \frametitle{Where do we see them?}

  \begin{itemize}
    \item \textbf{Computer Science:}
    \begin{itemize}
      \item Cellular Automata (discrete state, discrete time, discrete space)
      \item Network Dynamics (internet traffic, synchronization)
      \item Algorithmic Convergence
      \item Machine Learning: Weight updates in neural networks
    \end{itemize}
    \pause
    \item \textbf{Electrical \& Computer Engineering:}
    \begin{itemize}
      \item Circuits: RLC circuits ($\ddot{q} + R/L \dot{q} + 1/LC q = V(t)/L$)
      \item Control Systems (feedback loops)
      \item Communication Systems (synchronization, coding)
      \item Robotics, Power Grids
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{State Space and Trajectories}
  \frametitle{Visualizing Dynamics}

  \begin{itemize}
    \item The State Space is the set of all possible states the system can be in. Often $\R^n$.
    \item A trajectory (or orbit) is the sequence of states starting from an initial condition $x_0$: $\{ \Phi_t(x_0) \mid t \in T \}$.
    \item For continuous systems $\dot{\statev} = \vfield(\statev)$, trajectories are curves in state space tangent to the vector field $\vfield$.
    \item For discrete systems $\statev_{k+1} = \mathbf{f}(\statev_k)$, trajectories are sequences of points $x_0, x_1 = f(x_0), x_2 = f(f(x_0)), \dots$.
    \item \textbf{Phase Portrait:} A qualitative visualization of the dynamics in state space by showing a representative set of trajectories.
  \end{itemize}

  % \note{Speaker notes: Emphasize that the state space captures ALL necessary information to predict future behavior (assuming sufficient regularity conditions).}
\end{frame}

%=========================================================
\section{Linear Systems}
%=========================================================

\begin{frame}{Linear Systems (Continuous Time)}
  \frametitle{The Solvable Case: $\dot{\statev} = A\statev$}

  \begin{itemize}
    \item A system $\dot{\statev} = \vfield(\statev)$ is linear if $\vfield(\statev) = A\statev$ for some $n \times n$ matrix $A$.
    \item The solution with initial condition $\statev(0) = \statev_0$ is $\statev(t) = e^{At}\statev_0$.
    \item The behavior is determined by the eigenvalues $\lambda_i$ and eigenvectors $v_i$ of matrix $A$.
    \item If $A$ is diagonalizable, $e^{At} = V e^{\Lambda t} V^{-1}$, where $V$ columns are eigenvectors and $\Lambda$ is diagonal with eigenvalues.
    \item Trajectories are combinations of $e^{\lambda_i t} v_i$.
  \end{itemize}

  % \note{Speaker notes: Briefly recall matrix exponential properties. Focus on the connection between eigenvalues and dynamics.}
\end{frame}


\begin{frame}{Eigenvalues and Stability (Continuous Time)}
  \frametitle{Classification by Eigenvalues}

  For $\dot{\statev} = A\statev$, the origin $\statev=0$ is an equilibrium point. Its stability depends on $\text{Re}(\lambda_i)$:
  \begin{itemize}
    \item If $\text{Re}(\lambda_i) < 0$ for all $i$: {\color<2-3>{blue}Asymptotically Stable Node/Focus} (Sink)
    \item If $\text{Re}(\lambda_i) > 0$ for all $i$: {\color<2-3>{red}Unstable Node/Focus} (Source)
    \item If $\text{Re}(\lambda_i) = 0$ for all $i$: {\color<2-3>{green}Stable (but not Asymptotically Stable)} (Center - eigenvalues pure imaginary)
    \item If there are eigenvalues with $\text{Re}(\lambda_i) > 0$ and $\text{Re}(\lambda_j) < 0$: {\color<2-3>{orange}Saddle Point}
  \end{itemize}
  \pause
  Complex eigenvalues $\lambda \pm i\omega$ indicate spirals (foci) if $\text{Re}(\lambda) \neq 0$, or centers if $\text{Re}(\lambda) = 0$.

  % \note{Speaker notes: Mention the difference between node (real eigenvalues) and focus (complex eigenvalues). Show simple 2D examples.}
\end{frame}

\begin{frame}{Phase Portraits of 2D Linear Systems}
  \frametitle{Visualizing $\dot{\statev} = A\statev$}

  \begin{itemize}
    \item Node (Stable): $\lambda_1, \lambda_2 < 0$ (real)
    \item Focus (Stable): $\lambda, \bar{\lambda}$ with $\text{Re}(\lambda) < 0$ (complex conjugate)
    \item Saddle: $\lambda_1 < 0 < \lambda_2$ (real)
    \item Center: $\pm i\omega$ (pure imaginary)
  \end{itemize}

  % Placeholder for figures
  \begin{columns}[T]
    \column{0.48\textwidth}
      \begin{figure}
        \centering
        % \includegraphics[width=\textwidth]{stable_node_phase_portrait}
        Figure Placeholder: Stable Node Phase Portrait ($\lambda_1, \lambda_2 < 0$)
        \caption*{Stable Node}
      \end{figure}
    \column{0.48\textwidth}
      \begin{figure}
        \centering
        % \includegraphics[width=\textwidth]{saddle_phase_portrait}
        Figure Placeholder: Saddle Phase Portrait ($\lambda_1<0, \lambda_2>0$)
        \caption*{Saddle Point}
      \end{figure}
  \end{columns}
  % Add more figures for stable focus and center if desired.

  % \note{Speaker notes: Point out the flow direction, invariant subspaces (eigenvectors for real eigenvalues).}
\end{frame}


\begin{frame}{Linear Systems (Discrete Time)}
  \frametitle{Maps: $\statev_{k+1} = A\statev_k$}

  \begin{itemize}
    \item A system $\statev_{k+1} = \mathbf{f}(\statev_k)$ is linear if $\mathbf{f}(\statev_k) = A\statev_k$ plus perhaps a constant, which can be eliminated by shifting the origin if $A$ is invertible.
    \item The solution with initial condition $\statev_0$ is $\statev_k = A^k \statev_0$.
    \item The behavior is determined by the eigenvalues $\lambda_i$ and eigenvectors $v_i$ of matrix $A$.
    \item Trajectories are combinations of $\lambda_i^k v_i$.
  \end{itemize}

  % \note{Speaker notes: Comparison to continuous case. Matrix powers vs matrix exponential.}
\end{frame}

\begin{frame}{Eigenvalues and Stability (Discrete Time)}
  \frametitle{Classification by Eigenvalues for Maps}

  For $\statev_{k+1} = A\statev_k$, the origin $\statev=0$ is a fixed point. Its stability depends on $|\lambda_i|$:
  \begin{itemize}
    \item If $|\lambda_i| < 1$ for all $i$: {\color<2-3>{blue}Asymptotically Stable Node/Focus} (Sink)
    \item If $|\lambda_i| > 1$ for all $i$: {\color<2-3>{red}Unstable Node/Focus} (Source)
    \item If $|\lambda_i| = 1$ for all $i$: {\color<2-3>{green}Stable (non-hyperbolic)} (Can be tricky: periodic, quasi-periodic behavior on circles/tori)
    \item If there are eigenvalues with $|\lambda_i| < 1$ and $|\lambda_j| > 1$: {\color<2-3>{orange}Saddle Point}
  \end{itemize}
  \pause
  Real eigenvalues $\lambda$ with $|\lambda|<1$ give contraction; $|\lambda|>1$ expansion. Complex eigenvalues $\lambda = re^{i\theta}$ with $r \neq 0, 1$ give spirals.

  % \note{Speaker notes: Contrast with continuous time ($\text{Re}(\lambda)$ vs $|\lambda|$). Emphasize the boundary $|\lambda|=1$ is similar to $\text{Re}(\lambda)=0$.}
\end{frame}


%=========================================================
\section{Non-Linear Systems: Local Analysis}
%=========================================================

\begin{frame}{Introduction to Non-Linear Systems}
  \frametitle{The General Case}

  \begin{itemize}
    \item Systems not of the form $\dot{\statev} = A\statev$ or $\statev_{k+1} = A\statev_k$.
    \item $\dot{\statev} = \vfield(\statev, t)$ or $\statev_{k+1} = \mathbf{f}(\statev_k, k)$. (Often autonomous: $\vfield, \mathbf{f}$ don't depend explicitly on $t$ or $k$).
    \item Superposition principle does \alert{not} hold.
    \item Cannot generally find closed-form solutions.
    \item Analysis relies on qualitative and numerical methods.
  \end{itemize}
  \pause
  \textbf{Key Questions:}
  \begin{itemize}
    \item Where are the equilibrium points?
    \item What is the behavior near equilibrium points?
    \item What happens far from equilibrium points? (Global behavior)
    \item What happens as parameters change? (Bifurcations)
    \item Can the behavior be chaotic?
  \end{itemize}

  % \note{Speaker notes: Non-linearity is where the interesting, complex behavior emerges.}
\end{frame}

\begin{frame}{Equilibrium Points / Fixed Points}
  \frametitle{Points of No Change}

  \begin{definition}
    For $\dot{\statev} = \vfield(\statev)$, an equilibrium point $\statev^*$ is a state where $\vfield(\statev^*) = \mathbf{0}$. If $x_0 = \statev^*$, then $\statev(t) = \statev^*$ for all $t$.
  \end{definition}

  \begin{definition}
    For $\statev_{k+1} = \mathbf{f}(\statev_k)$, a fixed point $\statev^*$ is a state where $\statev^* = \mathbf{f}(\statev^*)$. If $\statev_0 = \statev^*$, then $\statev_k = \statev^*$ for all $k$.
  \end{definition}

  Finding equilibrium/fixed points requires solving algebraic equations: $\vfield(\statev) = \mathbf{0}$ or $\statev = \mathbf{f}(\statev)$.

  \begin{example}
    Logistic Equation: $\dot{N} = rN(1 - N/K)$. Equilibria: $rN(1-N/K)=0 \implies N^*=0, N^*=K$.
    Logistic Map: $x_{k+1} = rx_k(1-x_k)$. Fixed points: $x^* = rx^*(1-x^*) \implies x^*=0, x^* = (r-1)/r$.
  \end{example}

  % \note{Speaker notes: These are the simplest forms of invariant sets.}
\end{frame}


\begin{frame}{Linearization Around Equilibrium Points}
  \frametitle{Approximating Non-Linearity Locally}

  Consider $\dot{\statev} = \vfield(\statev)$. Let $\statev^*$ be an equilibrium point, $\vfield(\statev^*) = \mathbf{0}$.
  Let $\delta \statev = \statev - \statev^*$.
  $\dot{\delta \statev} = \dot{\statev} = \vfield(\statev^* + \delta \statev)$.
  Using a Taylor expansion around $\statev^*$:
  $\vfield(\statev^* + \delta \statev) \approx \vfield(\statev^*) + D\vfield(\statev^*) \delta \statev + O(\norm{\delta \statev}^2)$.
  Since $\vfield(\statev^*) = \mathbf{0}$, we get
  $\dot{\delta \statev} \approx D\vfield(\statev^*) \delta \statev$.

  Define $A = D\vfield(\statev^*)$, the Jacobian matrix of $\vfield$ evaluated at $\statev^*$.
  The linearized system is $\dot{\delta \statev} = A \delta \statev$.

  Similar principle applies to maps: $\mathbf{f}(\statev) \approx \mathbf{f}(\statev^*) + D\mathbf{f}(\statev^*) (\statev - \statev^*)$.
  For a fixed point $\statev^* = \mathbf{f}(\statev^*)$, let $\delta \statev_k = \statev_k - \statev^*$.
  $\statev_{k+1}^* + \delta \statev_{k+1} = \mathbf{f}(\statev^* + \delta \statev_k) \approx \mathbf{f}(\statev^*) + D\mathbf{f}(\statev^*) \delta \statev_k$.
  $\statev_{k+1}^* = \mathbf{f}(\statev^*)$, so $\delta \statev_{k+1} \approx D\mathbf{f}(\statev^*) \delta \statev_k$.
  Let $A = D\mathbf{f}(\statev^*)$. The linearized system is $\delta \statev_{k+1} = A \delta \statev_k$.

  % \note{Speaker notes: This is the key technique to analyze local behavior, leverage the known solutions of linear systems. Jacobian matrix calculation.}
\end{frame}

\begin{frame}{Hartman-Grobman Theorem}
  \frametitle{Relating Non-Linear and Linear Dynamics Locally}

  \begin{theorem}[Hartman-Grobman]
    Let $\statev^*$ be a hyperbolic equilibrium point of $\dot{\statev} = \vfield(\statev)$. This means that $A = D\vfield(\statev^*)$ has no eigenvalues with zero real part.
    Then there exists a neighborhood $U$ of $\statev^*$ where the flow of the non-linear system is topologically equivalent (homeomorphic) to the flow of the linearized system $\dot{\delta \statev} = A \delta \statev$.
  \end{theorem}

  \begin{itemize}
    \item Topological equivalence means there's a continuous, invertible map (homeomorphism) that maps trajectories of one system to trajectories of the other, preserving their direction.
    \item This theorem justifies using linearization to determine the qualitative behavior (like node, focus, saddle) near *hyperbolic* equilibrium points.
    \item It does \alert{not} apply if $\text{Re}(\lambda_i) = 0$ (non-hyperbolic points, e.g., centers in 2D linear systems). That's where bifurcations happen.
  \end{itemize}

  A similar theorem exists for discrete systems if $A=D\mathbf{f}(\statev^*)$ has no eigenvalues with magnitude 1.

  % \note{Speaker notes: Explain "topologically equivalent" – same qualitative picture, maybe stretched or distorted. Emphasize the hyperboilicity condition is crucial.}
\end{frame}

\begin{frame}{Lyapunov Stability}
  \frametitle{Rigorous Definition of Stability}

  Linearization works for hyperbolic points. How to define stability generally, or for non-hyperbolic points? Lyapunov's method.

  \begin{definition}
    An equilibrium point $\statev^*$ of $\dot{\statev} = \vfield(\statev)$ is:
    \begin{itemize}
      \item \textbf{Lyapunov Stable} if for every neighborhood $U$ of $\statev^*$, there exists a neighborhood $V \subset U$ of $\statev^*$ such that any trajectory starting in $V$ remains in $U$ for all future time $t>0$. (Trajectories starting close stay close).
      \item \textbf{Asymptotically Stable} if it is Lyapunov stable and $\lim_{t\to\infty} \statev(t) = \statev^*$ for all initial conditions $\statev(0)$ in some neighborhood of $\statev^*$. (Trajectories starting close converge to $\statev^*$).
      \item \textbf{Unstable} if it is not Lyapunov stable.
    \end{itemize}
  \end{definition}
  (Similar definitions apply to fixed points of maps for discrete time).

  \begin{itemize}
    \item Asymptotic stability for linear systems corresponds to all eigenvalues having $\text{Re}(\lambda)<0$ (continuous) or $|\lambda|<1$ (discrete).
    \item Lyapunov stability (but not asymptotic) corresponds to $\text{Re}(\lambda)\leq 0$ with some $\text{Re}(\lambda)=0$ (continuous) or $|\lambda|\leq 1$ with some $|\lambda|=1$ (discrete), *and* those on the boundary are simple eigenvalues for diagonalizable matrices. Non-linear terms can destabilize these!
  \end{itemize}

  % \note{Speaker notes: Contrast Lyapunov vs Asymptotic. Lyapunov is about staying within a region, Asymptotic is about converging to the point.}
\end{frame}

\begin{frame}{Lyapunov Functions}
  \frametitle{A Method to Prove Stability/Instability}

  \begin{definition}
    A scalar function $V(\statev)$ is a Lyapunov function for $\dot{\statev} = \vfield(\statev)$ at an equilibrium $\statev^*$ if, in a neighborhood of $\statev^*$:
    \begin{enumerate}
      \item $V(\statev^*) = 0$ and $V(\statev) > 0$ for $\statev \neq \statev^*$ (Positive definite).
      \item The orbital derivative $\dot{V}(\statev) = \nabla V(\statev) \cdot \vfield(\statev)$ satisfies:
      \begin{itemize}
        \item $\dot{V}(\statev) \leq 0$: $\statev^*$ is Lyapunov stable.
        \item $\dot{V}(\statev) < 0$ (for $\statev \neq \statev^*$): $\statev^*$ is asymptotically stable.
        \item $\dot{V}(\statev) > 0$ (for $\statev \neq \statev^*$): $\statev^*$ is unstable (if $V$ can also be negative somewhere).
      \end{itemize}
    \end{enumerate}
  \end{definition}

  \begin{itemize}
    \item Finding a Lyapunov function can be hard, but if found, it provides a rigorous stability proof without integrating trajectories.
    \item Represents a "energy-like" quantity that decreases along trajectories.
    \item \textbf{LaSalle's Invariance Principle:} If $\dot{V}(\statev) \leq 0$, trajectories converge to the largest invariant set inside $\{\statev \mid \dot{V}(\statev)=0\}$. Useful when $\dot{V}$ is only negative semi-definite.
  \end{itemize}

  % \note{Speaker notes: This tool is very important in control theory and analysis of complex systems where linearization isn't enough or doesn't apply (e.g., non-hyperbolic points).}
\end{frame}


%=========================================================
\section{Global Analysis: Flows, Maps, Limit Sets, Attractors}
%=========================================================

\begin{frame}{Flows and Maps}
  \frametitle{Formal Definitions of Time Evolution}

  \begin{definition}[Flow]
    A flow on a space $X$ is a map $\Phi: \R \times X \to X$ such that:
    \begin{enumerate}
      \item $\Phi(0, x) = x$ for all $x \in X$ (Identity at $t=0$).
      \item $\Phi(t, \Phi(s, x)) = \Phi(t+s, x)$ for all $t, s \in \R$, $x \in X$ (Semigroup/Group property).
    \end{enumerate}
    If $\Phi$ is continuous/differentiable with respect to $t$ and $x$, it's the solution to $\dot{x} = \vfield(x)$. $\Phi_t(x) = \Phi(t, x)$.
  \end{definition}

  \begin{definition}[Map/Cascade]
    A map on a space $X$ is a single function $f: X \to X$. The discrete dynamical system is generated by iteration: $\Phi_k(x) = f^k(x)$.
  \end{definition}

  \begin{itemize}
    \item Non-autonomous systems $\dot{x} = F(x,t)$ or $x_{k+1} = f(x_k, k)$ don't strictly define a flow/map on state space $X$, but can sometimes be made autonomous by augmenting the state space (e.g., including $t$ as a state variable).
    \item Global analysis looks at the behavior of trajectories as $t \to \pm\infty$.
  \end{itemize}

  % \note{Speaker notes: Flow is continuous time evolution, governed by ODEs. Map is discrete time evolution, from difference equations or Poincaré section.}
\end{frame}


\begin{longframe} % Use longframe to allow breaking
  \frametitle{Limit Sets}

  What is the ultimate fate of a trajectory as $t \to \infty$ or $k \to \infty$?

  \begin{definition}[Omega Limit Set $\omega(x_0)$]
    For a trajectory $\gamma(t) = \Phi_t(x_0)$, the $\omega$-limit set is the set of points that the trajectory approaches as $t \to \infty$:
    $\omega(x_0) = \{ y \in X \mid \exists \text{ sequence } t_n \to \infty \text{ such that } \Phi_{t_n}(x_0) \to y \}$.
  \end{definition}

  \begin{definition}[Alpha Limit Set $\alpha(x_0)$]
    Similarly, for $t \to -\infty$:
    $\alpha(x_0) = \{ y \in X \mid \exists \text{ sequence} t_n \to -\infty \text{ such that } \Phi_{t_n}(x_0) \to y \}$.
  \end{definition}

  \begin{remark}
    For discrete systems $\statev_{k+1} = \mathbf{f}(\statev_k)$, limit sets $\omega(x_0)$ are defined by sequences $k_n \to \infty$.
  \end{remark}


  \begin{proposition}
    If a trajectory $\Phi_t(x_0)$ is bounded, its $\omega$-limit set $\omega(x_0)$ is non-empty, compact, and invariant.
  \end{proposition}

  \begin{definition}[Invariant Set]
    A set $M \subset X$ is invariant under $\Phi_t$ if $\Phi_t(M) = M$ for all $t \in T$.
  \end{definition}

  \begin{itemize}
    \item Equilibrium points points are invariant sets.
    \item Periodic orbits are invariant sets.
    \item Limit sets are always invariant (if trajectory is bounded).
  \end{itemize}

  % \note{Speaker notes: The limit set is where the trajectory "ends up". It must be invariant because if a point y in the limit set could leave over time t, then the sequence t_n approaching y could also leave, contradicting y being a limit point.}
\end{longframe}


\begin{frame}{Attractors}
  \frametitle{Sets That Attract Neighboring Trajectories}

  Limit sets describe the fate of a single trajectory. An attractor is a set that *attracts* an open set of initial conditions. Definition variations exist!

  \begin{definition}[Common Definition of Attractor]
    A set $A \subset X$ is an attractor if:
    \begin{enumerate}
      \item $A$ is invariant.
      \item There is a neighborhood $U$ of $A$ (its basin of attraction) such that for any $x_0 \in U$, $\omega(x_0) \subset A$.
      \item There is no proper subset of $A$ that satisfies (1) and (2). (Minimality/Indecomposability)
      \item Often, $A$ is required to be a limit set itself ($A = \omega(x)$ for most $x$ in its basin) or some form of mixing/transitivity.
    \end{enumerate}
  \end{definition}

  \begin{itemize}
    \item \textbf{Examples:} Stable equilibrium points, stable limit cycles (periodic orbits), strange attractors.
    \item The \textbf{Basin of Attraction} $B(A)$ of an attractor $A$ is the set of all initial conditions $x_0$ such that $\omega(x_0) \subset A$.
    \item Basins of attraction can have complex, fractal boundaries in non-linear systems.
  \end{itemize}

  % \note{Speaker notes: Attractors are the structures we observe the system settling into. Basins partition the state space according to which attractor a trajectory goes to.}
\end{frame}


\begin{frame}{Periodic Orbits and Limit Cycles}
  \frametitle{Recurring States}

  \begin{definition}[Periodic Orbit]
    A trajectory $\gamma(t) = \Phi_t(x_0)$ is periodic with period $T_p > 0$ if $\Phi_{T_p}(x_0) = x_0$.
    For discrete systems $\statev_{k+1} = f(\statev_k)$, a periodic orbit of period $p > 0$ is sequence $x_0, x_1, \dots, x_{p-1}$ such that $x_p = f^p(x_0) = x_0$ and $f^k(x_0) \neq x_0$ for $1 \leq k < p$.
  \end{definition}

  \begin{definition}[Limit Cycle]
    For continuous systems, a limit cycle is an isolated periodic orbit. Isolated means there is a neighborhood around the cycle that contains no other periodic orbits.
    Limit cycles can be stable (attracting) or unstable (repelling).
  \end{definition}

  \begin{itemize}
    \item Equilibrium points are period-0 orbits (degenerate).
    \item Linear systems can have families of periodic orbits (centers). Non-linear systems more typically have isolated limit cycles.
    \item Poincaré-Bendixson Theorem: Provides conditions for the existence of a limit cycle in 2D systems.
  \end{itemize}

  % \note{Speaker notes: Periodic orbits are dynamics looping back on themselves. Limit cycles are important non-equilibrium attractors.}
\end{frame}

\begin{frame}{Poincaré-Bendixson Theorem}
  \frametitle{Limit Set Structure in 2D}

  \begin{theorem}[Poincaré-Bendixson Theorem]
    Consider a continuous dynamical system $\dot{\statev} = \vfield(\statev)$ in $\R^2$. If a trajectory $\statev(t)$ is bounded for $t \ge 0$ and does not approach any equilibrium point as $t \to \infty$, then its $\omega$-limit set is a periodic orbit.
  \end{theorem}

  \begin{itemize}
    \item This theorem implies that in 2D, bounded non-equilibrium limit sets must be periodic orbits.
    \item It rules out chaotic attractors or quasiperiodic motion as $\omega$-limit sets in 2D continuous systems.
    \item Does \alert{not} apply in $\R^n$ for $n \geq 3$.
    \item Does \alert{not} apply to discrete systems.
  \end{itemize}

  \begin{example}
    Van der Pol Oscillator: $\ddot{x} - \mu(1-x^2)\dot{x} + x = 0$. For $\mu > 0$, rewrite as system in $\R^2$. It has a limit cycle.
  \end{example}

  % Placeholder for figure
  \begin{figure}
    \centering
    % \includegraphics[width=0.6\textwidth]{vanderpol_limit_cycle}
    Figure Placeholder: Van der Pol Limit Cycle Phase Portrait
    \caption*{Limit cycle in Van der Pol system for $\mu > 0$}
  \end{figure}

  % \note{Speaker notes: A very powerful theorem, specific to 2D continuous dynamics. Highlights that complex limit sets require higher dimensions or discrete time.}
\end{frame}


%=========================================================
\section{Bifurcations}
%=========================================================

\begin{frame}{What is a Bifurcation?}
  \frametitle{Qualitative Changes via Parameter}

  \begin{definition}
    A bifurcation occurs when a small smooth change in a system parameter causes a sudden "qualitative" or "topological" change in its dynamics.
  \end{definition}

  \begin{itemize}
    \item Qualitative changes include: change in number/stability of equilibrium points/periodic orbits, birth/death of invariant sets, change in dimension of attractors.
    \item Bifurcations typically occur when equilibrium points (or periodic orbits) change their stability type. For equilibrium points, this happens when eigenvalues of the linearization cross the stability boundary:
    \begin{itemize}
      \item Continuous: $\text{Re}(\lambda)$ crosses zero.
      \item Discrete: $|\lambda|$ crosses one.
    \end{itemize}
    \item The parameter value where the change happens is a \textbf{bifurcation point}.
  \end{itemize}

  \begin{figure}
    \centering
    % \includegraphics[width=0.7\textwidth]{bifurcation_diagram_concept}
    Figure Placeholder: Conceptual Bifurcation Diagram (Parameter vs State)
    \caption*{Conceptual Bifurcation Diagram}
  \end{figure}

  % \note{Speaker notes: Bifurcations are crucial for understanding how system behavior changes with context (parameters). They are points of critical transition.}
\end{frame}

\begin{frame}{Codimension-1 Bifurcations (Continuous)}
  \frametitle{Common Bifurcations in $\dot{x} = f(x, r)$}

  These occur when a single eigenvalue passes through zero. (Assume $x \in \R^1$ or focus on dynamics on a center manifold).

  \begin{itemize}
    \item \textbf{Saddle-Node (or Fold) Bifurcation:} Two equilibria (one stable, one unstable) collide and annihilate as parameter $r$ changes.
    Equation example: $\dot{x} = r - x^2$. Equilibria $x^* = \pm \sqrt{r}$ for $r>0$. None for $r<0$. At $r=0$, $x^*=0$ is a non-hyperbolic equilibrium.
    \begin{figure}
      \centering
      % \includegraphics[width=0.45\textwidth]{saddle_node_bifurcation_diagram}
      Figure Placeholder: Saddle-Node Bifurcation Diagram ($r$ vs $x^*$)
      \caption*{Saddle-Node Bifurcation}
    \end{figure}

    \item \textbf{Transcritical Bifurcation:} Two equilibria exist for all $r$, they collide and exchange stability as $r$ changes.
    Equation example: $\dot{x} = rx - x^2$. Equilibria $x^*=0, x^*=r$. At $r=0$, they collide.

    \item \textbf{Pitchfork Bifurcation:} A single equilibrium loses stability and gives rise to two new stable (supercritical) or unstable (subcritical) equilibria due to a symmetry in the system.
    Equation example: $\dot{x} = rx - x^3$. Equilibria $x^*=0, x^*=\pm \sqrt{r}$ for $r>0$. At $r=0$, $x^*=0$ changes stability. (Supercritical shown).
  \end{itemize}
\end{frame}

\begin{frame}{Understanding Pitchfork Bifurcations}
    \frametitle{Pitchfork Bifurcation ($\dot{x} = rx - x^3$)}

    \begin{itemize}
        \item At $r < 0$: $x^*=0$ is the only equilibrium, and it is stable ($\diff{f}{x}|_0 = r < 0$).
        \item At $r = 0$: $x^*=0$ is non-hyperbolic ($\diff{f}{x}|_0 = 0$). A transcritical bifurcation happens locally at $(r, x) = (0, 0)$ if you rewrite carefully.
        \item At $r > 0$: $x^*=0$ becomes unstable ($\diff{f}{x}|_0 = r > 0$). Two new stable equilibria appear at $x^* = \pm \sqrt{r}$.
        \item This structure only requires odd powers of $x$ due to $\alert{Z_2 \text{ symmetry}}$ (system is invariant under $x \to -x$). $\dot{-x} = r(-x) - (-x)^3 = -rx + x^3 = -(rx - x^3) = -\dot{x}$.
    \end{itemize}

    \begin{columns}[T]
        \column{0.48\textwidth}
        \begin{figure}
            \centering
            % \includegraphics[width=\textwidth]{pitchfork_bifurcation_diagram}
            Figure Placeholder: Supercritical Pitchfork Diagram
            \caption*{Supercritical Pitchfork}
        \end{figure}
        \column{0.48\textwidth}
        \begin{figure}
            \centering
            % \includegraphics[width=\textwidth]{subcritical_pitchfork_bifurcation_diagram}
            Figure Placeholder: Subcritical Pitchfork Diagram
            \caption*{Subcritical Pitchfork ($\dot{x} = rx + x^3$)}
        \end{figure}
    \end{columns}

    % \note{Speaker notes: Explain difference between super- and subcritical - new eq are stable/unstable respectively. Subcritical can lead to hysteresis and sudden jumps.}
\end{frame}

\begin{frame}{Hopf Bifurcation}
  \frametitle{Birth of a Limit Cycle}

  Occurs when a pair of complex conjugate eigenvalues crosses the imaginary axis. (From $\text{Re}(\lambda) < 0$ to $\text{Re}(\lambda) > 0$).

  \begin{itemize}
    \item Typically occurs in systems of dimension $n \ge 2$.
    \item A stable equilibrium loses stability and gives rise to a limit cycle.
    \item \textbf{Supercritical Hopf:} Stable equilibrium becomes unstable, a small \alert{stable} limit cycle appears around it.
    \item \textbf{Subcritical Hopf:} Stable equilibrium becomes unstable, a small \alert{unstable} limit cycle exists *before* the bifurcation point, which merges with the equilibrium at the bifurcation. This can lead to sudden jumps in amplitude.
  \end{itemize}

  \begin{figure}
    \centering
    % \includegraphics[width=0.6\textwidth]{supercritical_hopf_diagram}
    Figure Placeholder: Supercritical Hopf Bifurcation in Phase Space and Parameter Space
    \caption*{Supercritical Hopf Bifurcation (Phase space view as parameter changes across zero)}
  \end{figure}

  \begin{itemize}
    \item The limit cycle frequency is approximately the imaginary part of the eigenvalues at the bifurcation.
    \item Very common in systems that start oscillating as a parameter is increased (e.g., circuits, chemical reactions, fluid dynamics).
  \end{itemize}

  % \note{Speaker notes: This is the primary mechanism for periodic oscillations to appear in non-linear systems. Link to Van der Pol previous example.}
\end{frame}

\begin{frame}{Bifurcations in Discrete Systems}
  \frametitle{Maps: $x_{k+1} = f(x_k, r)$}

  Fixed points $x^*$ where $x^* = f(x^*, r)$ change stability when the eigenvalue of the linearization (which is $f'(x^*, r)$ for 1D systems) crosses magnitude 1.

  \begin{itemize}
    \item \textbf{Saddle-Node Bifurcation:} A pair of fixed points created/destroyed as $f'(x^*)=1$. Analogous to continuous case.
    \item \textbf{Transcritical Bifurcation:} Fixed points collide and exchange stability as $f'(x^*)=1$. Analogous.
    \item \textbf{Pitchfork Bifurcation:} A fixed point loses stability and two new ones appear as $f'(x^*)=-1$. Occurs for symmetric maps.
    \item \textbf{Period-Doubling (Flip) Bifurcation:} A fixed point loses stability as $f'(x^*)$ passes through \alert{-1}. A stable period-2 orbit is born. This doesn't have a direct analogue in continuous time (no eigenvalue -1).
  \end{itemize}

  Discrete bifurcations are key to understanding the transition to chaos in maps (e.g., the Logistic Map).

  % \note{Speaker notes: Highlight the $|\lambda|=1$ boundary condition. Period-doubling is unique to discrete systems and fundamental route to chaos.}
\end{frame}


%=========================================================
\section{Chaos}
%=========================================================

\begin{frame}{Introduction to Chaotic Dynamics}
  \frametitle{Deterministic ... but Unpredictable?}

  \begin{itemize}
    \item Chaotic systems are deterministic first-order systems (no explicit randomness) that exhibit seemingly random, unpredictable behavior.
    \item Key properties:
    \begin{enumerate}
      \item \textbf{Sensitive Dependence on Initial Conditions:} Nearby trajectories separate exponentially fast. ("Butterfly Effect")
      \item \textbf{Topological Mixing:} Regions in state space get mixed by the dynamics.
      \item \textbf{Dense Periodic Orbits:} Periodic orbits are scattered throughout the chaotic region (though most initial conditions are not periodic).
    \end{enumerate}
    \item Chaos typically occurs in:
    \begin{itemize}
      \item Continuous systems with state space dimension $n \ge 3$.
      \item Discrete systems with state space dimension $n \ge 1$.
    \end{itemize}
    \item Chaotic behavior occurs on a specific invariant set called a \alert{Strange Attractor}.
  \end{itemize}

  % \note{Speaker notes: Correct common misconceptions: Chaos is not randomness, it's deterministic complexity. Not just complex behavior, specific properties make it chaotic.}
\end{frame}

\begin{frame}{Sensitive Dependence on Initial Conditions}
  \frametitle{The Butterfly Effect}

  Let $\statev_1(t)$ and $\statev_2(t)$ be two trajectories with very close initial conditions $\statev_1(0)$ and $\statev_2(0)$, small distance $\norm{\delta \statev(0)} = \norm{\statev_1(0) - \statev_2(0)}$.
  For chaotic systems, this distance typically grows exponentially for some finite time:
  $\norm{\delta \statev(t)} \approx \norm{\delta \statev(0)} e^{\lambda t}$ for $\lambda > 0$.

  \begin{itemize}
    \item $\lambda$ is related to the \alert{Lyapunov exponent}.
    \item This implies predictive power is limited to a finite time horizon $T \sim \frac{1}{\lambda} \log \left(\frac{\text{size of attractor}}{\text{initial error}}\right)$.
    \item Even with infinitesimal initial error, the trajectory becomes unpredictable beyond a certain point.
  \end{itemize}

  % Placeholder for figure
  \begin{figure}
    \centering
    % \includegraphics[width=0.6\textwidth]{lyapunov_divergence_plot}
    Figure Placeholder: Exponential Divergence of Nearby Trajectories
    \caption*{Sensitive Dependence}
  \end{figure}

  % \note{Speaker notes: This exponential separation is the defining characteristic from a practical prediction point of view.}
\end{frame}

\begin{frame}{Lyapunov Exponents}
  \frametitle{Quantifying Exponential Growth/Shrinkage}

  Lyapunov exponents measure the average rate of exponential divergence or convergence of nearby trajectories. For an $n$-dimensional system, there are $n$ Lyapunov exponents $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$.

  \begin{itemize}
    \item Consider an infinitesimal sphere of initial conditions. Over time, this sphere deforms into an ellipsoid. The Lyapunov exponents are related to the principal axes' exponential growth/shrinkage rates.
    \item For $\dot{\statev} = \vfield(\statev)$, the exponents are related to the eigenvalues of the linearized flow map $D\Phi_t$: $\norm{D\Phi_t v} \sim e^{\lambda t} \norm{v}$.
    \item A system is defined as \alert{chaotic} if at least one Lyapunov exponent is positive ($\lambda_1 > 0$).
    \item The sum of Lyapunov exponents $\sum \lambda_i$ relates to the rate of volume contraction in state space ($\approx \text{tr}(D\vfield)$ for continuous systems).
    \item Dissipative systems have $\sum \lambda_i < 0$. Strange attractors are found in dissipative systems.
  \end{itemize}

  \begin{example}
    Lorenz system $\left(\sum \lambda_i < 0\right)$: $(\lambda_1, \lambda_2, \lambda_3) \approx (0.9, 0, -14.6)$
    Chaos because $\lambda_1 > 0$. One exponent is zero ($\lambda_2=0$) corresponding to evolution along the trajectory itself. Sum is negative: dissipative.
  \end{example}

  % \note{Speaker notes: Lyapunov exponents provide a concrete numerical measure of chaos. Positive exponent = sensitive dependence.}
\end{frame}

\begin{frame}{Poincaré Sections}
  \frametitle{Reducing Dimension, Revealing Structure}

  A technique to analyze the dynamics of a flow by looking at its intersection with a lower-dimensional hyperplane (surface of section) in state space.

  \begin{itemize}
    \item Choose a $(n-1)$-dimensional surface $\Sigma$ transversal to the flow (trajectories ideally cross it, not tangent).
    \item Track the points where a trajectory intersects $\Sigma$ sequentially: $x_0' \in \Sigma$, $x_1' = \Phi_{T_1}(x_0') \in \Sigma$, $x_2' = \Phi_{T_2}(x_1') \in \Sigma$, etc. This defines a discrete map $P: \Sigma \to \Sigma$, the \alert{Poincaré map}.
    \item Periodic orbit of the flow $\leftrightarrow$ Fixed point of the Poincaré map.
    \item Limit cycle $\leftrightarrow$ Fixed point of the Poincaré map.
    \item Torus (quasiperiodic flow) $\leftrightarrow$ Circle of points on $\Sigma$.
    \item Chaotic attractor $\leftrightarrow$ A set of points on $\Sigma$ that often reveals fractal structure.
  \end{itemize}

  \begin{figure}
    \centering
    % \includegraphics[width=0.6\textwidth]{poincare_section_concept}
    Figure Placeholder: Concept of Poincaré Section
    \caption*{Poincaré Section: from continuous trajectory to discrete points on a lower-dim manifold}
  \end{figure}

  % \note{Speaker notes: This transforms a continuous system study into a discrete map study, which can be simpler. It's the standard computational tool to "see" attractors in 3D+ systems or verify periodicity.}
</frame>

\begin{frame}{Strange Attractors}
  \frametitle{Attractors with Fractal Structure}

  \begin{definition}
    A strange attractor is an attractor that has a fractal structure and exhibits sensitive dependence on initial conditions (i.e., it's a chaotic attractor).
  \end{definition}

  \begin{itemize}
    \item Fractal structure: self-similar on different scales, non-integer dimension (e.g., box-counting dimension, correlation dimension).
    \item Chaotic dynamics (positive Lyapunov exponent) typically occur on strange attractors for dissipative systems.
    \item Examples: Lorenz attractor, Hénon attractor (for a map), Rössler attractor.
  \end{itemize}

  % Placeholder for figures
  \begin{columns}[T]
    \column{0.48\textwidth}
      \begin{figure}
        \centering
        % \includegraphics[width=\textwidth]{lorenz_attractor}
        Figure Placeholder: Lorenz Attractor
        \caption*{Lorenz Attractor ($\beta=8/3, \rho=28, \sigma=10$)}
      \end{figure}
    \column{0.48\textwidth}
      \begin{figure}
        \centering
        % \includegraphics[width=\textwidth]{henon_attractor}
        Figure Placeholder: Hénon Attractor ($x_{n+1} = 1 - ax_n^2 + y_n, y_{n+1} = bx_n$)
        \caption*{Hénon Attractor ($a=1.4, b=0.3$)}
      \end{figure}
  \end{columns}

  % \note{Speaker notes: Show the characteristic shapes. Explain that the fractal structure is a consequence of stretching and folding dynamics characteristic of chaos.}
</frame>

%=========================================================
\section{Discrete Dynamical Systems (Maps)}
%=========================================================

\begin{frame}{Iterated Maps}
  \frametitle{Recurrence Relations $\statev_{k+1} = \mathbf{f}(\statev_k)$}

  Covered linear maps, but non-linear maps are just as rich, or richer, than flows, especially in demonstrating transitions to chaos.

  \begin{itemize}
    \item Iteration: $\statev_0, \statev_1 = \mathbf{f}(\statev_0), \statev_2 = \mathbf{f}(\mathbf{f}(\statev_0)) = \mathbf{f}^2(\statev_0), \dots, \statev_k = \mathbf{f}^k(\statev_0)$.
    \item Trajectories are sequences of points.
    \item Fixed points: $\statev^* = \mathbf{f}(\statev^*)$. (Period 1 orbits).
    \item Periodic orbits of period $p$: $\statev^* = \mathbf{f}^p(\statev^*)$ but $\statev^* \neq \mathbf{f}^k(\statev^*)$ for $1 \leq k < p$. Fixed points of $\mathbf{f}^p$.
    \item Stability of fixed point $\mathbf{x}^*$: linearized map $\delta \statev_{k+1} = D\mathbf{f}(\mathbf{x}^*) \delta \statev_k$. Stable if $|\lambda| < 1$ for all eigenvalues of $D\mathbf{f}(\mathbf{x}^*)$.

    % \note{Speaker notes: Maps are fundamental in CS (algorithms), ECE (digital systems), and used in physics/math (Poincare maps). Iteration is the core operation.}
  \end{itemize}
\end{frame}

\begin{frame}{Visualizing 1D Maps: Cobweb Plots}
  \frametitle{Graphical Analysis of $x_{k+1} = f(x_k)$}

  Useful for visualizing dynamics of 1D maps.

  \begin{enumerate}
    \item Plot the function $y = f(x)$ and the line $y = x$.
    \item Start at $x_0$ on the x-axis.
    \item Go vertically to the curve $y = f(x)$ to find $x_1 = f(x_0)$ (height).
    \item Go horizontally to the line $y = x$. The x-coordinate is now $x_1$.
    \item Repeat from step 2, starting at $x_1$.
  \end{enumerate}

  \begin{itemize}
    \item Fixed points are intersections of $y=f(x)$ and $y=x$.
    \item Stable fixed points pull the cobweb inward.
    \item Unstable fixed points push the cobweb outward.
    \item Periodic orbits create a "rectangle" or sequence of points.
    \item Chaos looks like the sequence of points filling an interval.
  \end{itemize}

  % Placeholder for figure
  \begin{figure}
    \centering
    % \includegraphics[width=0.6\textwidth]{cobweb_plot_example}
    Figure Placeholder: Cobweb Plot illustrating approach to fixed point or limit cycle
    \caption*{Cobweb Plot example}
  \end{figure}

  % \note{Speaker notes: Show how different dynamics look on a cobweb plot (convergence, divergence, 2-cycle, chaos).}
\end{frame>

\begin{frame}{The Logistic Map}
  \frametitle{A Case Study in $1D$ Chaos}

  The Iterated Logistic Equation: $x_{k+1} = rx_k(1-x_k)$, where $x_k \in [0, 1]$ (population normalized) and $r \in [0, 4]$ (growth parameter).

  \begin{itemize}
    \item Simple non-linear 1D map.
    \item Fixed points: $x^*=0$ and $x^*=(r-1)/r$.
    \item Bifurcation as $r$ increases:
    \begin{itemize}
      \item $0 \le r < 1$: $x^*=0$ is stable. Population dies out.
      \item $1 \le r < 3$: $x^*=(r-1)/r$ is stable (after transcritical at $r=1$). $x^*=0$ unstable. Population -> constant.
      \item $r = 3$: $x^* = (3-1)/3 = 2/3$. $f'(2/3, 3) = 3(1 - 2*2/3) = 3(1 - 4/3) = -1$. Period-doubling bifurcation!
      \item $3 < r < 3.449...$: $x^* $ is unstable. A stable period-2 cycle appears. $f^2(x)$ has stable fixed points.
      \item Sequence of period-doubling bifurcations as r increases: $2 \to 4 \to 8 \to 16 \dots$ at parameters $r_1 < r_2 < r_3 < \dots$.
    \end{itemize}
  \end{itemize}

  % \note{Speaker notes: A classic example showing complex behavior from simple equations. Connect back to 1D bifurcations.}
\end{frame}

\begin{frame}{The Period-Doubling Route to Chaos}
  \frametitle{Feigenbaum Universality}

  \begin{itemize}
    \item The sequence of period-doubling bifurcations $r_1 < r_2 < r_3 < \dots$ converges geometrically:
    $\lim_{n \to \infty} \frac{r_{n+1} - r_n}{r_n - r_{n-1}} = \delta \approx 4.6692$. (\alert{Feigenbaum constant $\delta$}).
    \item Similarly, the scaling of the intervals between branches in the bifurcation diagram shrinks geometrically by a factor $\alpha \approx 2.5029$.
    \item For $r > r_\infty \approx 3.5699$: Chaotic behavior appears. There are also periodic windows within the chaotic region.
    \item \alert{Universality:} These Feigenbaum constants $\delta$ and $\alpha$ are the \alert{same} for a large class of 1D maps with a single quadratic maximum ($f(x)=ax^2+bx+c$, $rx(1-x)$, etc., scaled appropriately). This was a major discovery showing structure underlying chaos.
  \end{itemize}

  % Placeholder for figure
  \begin{figure}
    \centering
    % \includegraphics[width=0.8\textwidth]{logistic_map_bifurcation_diagram}
    Figure Placeholder: Bifurcation Diagram of the Logistic Map ($r$ vs Long-term $x_k$)
    \caption*{Bifurcation Diagram of the Logistic Map}
  \end{figure}

  % \note{Speaker notes: This universality implies that the scaling behavior near the onset of chaos via this route is independent of the specific system details. Explain how to read the bifurcation diagram.}
\end{frame>

%=========================================================
\section{Computational Aspects and Applications}
%=========================================================

\begin{frame}{Numerical Simulation}
  \frametitle{Approximating Trajectories}

  Most non-linear systems lack analytical solutions, so numerical methods are essential.

  \begin{itemize}
    \item \textbf{Continuous Systems $\dot{\statev} = \vfield(\statev)$:}
    \begin{itemize}
      \item Euler method (simplest): $\statev(t + \Delta t) \approx \statev(t) + \vfield(\statev(t)) \Delta t$.
      \item Runge-Kutta methods (more accurate, e.g., RK4): involve evaluating $\vfield$ at multiple points within $\Delta t$.
      \item Implicit methods, adaptive step size methods...
      \item Caution: Numerical errors can accumulate, especially near unstable structures or when simulating chaotic systems (sensitive dependence!). Finite precision limits prediction time.
    \end{itemize}
    \item \textbf{Discrete Systems $\statev_{k+1} = \mathbf{f}(\statev_k)$:}
    \begin{itemize}
      \item Straightforward iteration $\statev_{k+1} = \mathbf{f}(\statev_k)$ is the simulation.
      \item Fixed point finding (e.g., Newton's method for $\mathbf{f}(\statev)-\statev=0$).
    \end{itemize}
  \end{itemize}

  % \note{Speaker notes: Numerical methods are the workhorse of dynamical systems research. Emphasize the challenges of chaos simulation - can't predict infinitely far.}
\end{frame}

\begin{frame}{From Data to Dynamics}
  \frametitle{Reconstructing State Space}

  What if we don't have equations, only time series data $y(t)$ or $y_k$?

  \begin{itemize}
    \item \textbf{Phase Space Reconstruction (Takens' Theorem):} For a sufficiently generic chaotic system and a single measurement variable $y(t)$, you can reconstruct a space topologically equivalent to the original strange attractor using time-delayed copies of the data:
    $Y(t) = (y(t), y(t+\tau), y(t+2\tau), \dots, y(t+(m-1)\tau))$.
    $m$ is the embedding dimension ($m \ge 2 \times D_{Attractor}$, where $D_{Attractor}$ is the box-counting or correlation dimension). $\tau$ is the delay time.
    \item This allows studying the properties (dimensions, Lyapunov exponents) of the underlying dynamics from data.
    \item \textbf{Data-driven Dynamical Systems:} Modern methods like Sparse Identification of Nonlinear Dynamics (SINDy) or Neural ODEs attempt to learn the governing equations $\dot{\statev} = \vfield(\statev)$ directly from data.
  \end{itemize}

  % \note{Speaker notes: This is a key link to experimental data and machine learning. Takens' Theorem provides theoretical justification for common data analysis techniques.}
\end{frame}

<frame}{Diverse Applications}
  \frametitle{Systems Thinking in Action}

  \begin{itemize}
    \item \textbf{Physics:} Weather forecasting, climate models, turbulence, laser physics, synchronized oscillators, complex materials.
    \item \textbf{Engineering:} Control system design, circuit design (non-linear oscillators), secure communication using chaos, network synchronization (power grids, communication networks), robotics.
    \item \textbf{Biology:} Population dynamics (predator-prey cycles, epidemics), neuroscience (neuron firing patterns, brain activity networks), cardiac rhythms, genetic regulatory networks.
    \item \textbf{Economics/Social Sciences:} Financial market models, spread of information/disease, opinion dynamics, game theory.
    \item \textbf{Computer Science:} Pseudorandom number generation, optimization algorithms (simulated annealing), analysis of algorithms, network traffic modeling, cellular automata, machine learning dynamics.
  \end{itemize}
  \pause
  Understanding dynamics provides a universal language and set of tools to analyze change in complex systems across disciplines.

  % \note{Speaker notes: Reinforce the interdisciplinary nature of the field. Encourage students to think about dynamical systems in their own research areas.}
\end{frame}

%=========================================================
\section{Conclusion}
%=========================================================

\begin{frame}{Summary}
  \frametitle{Key Concepts}

  \begin{itemize}
    \item Dynamical systems describe evolution over time: continuous (flows/ODEs) or discrete (maps).
    \item Linear systems are solvable; behavior determined by eigenvalues ($\text{Re}(\lambda)$ or $|\lambda|$). Lead to simple limit sets (equilibria, periodic orbits).
    \item Non-linear systems require different tools:
    \begin{itemize}
      \item Local: Equilibria, linearization, Hartman-Grobman, Lyapunov stability.
      \item Global: Limit sets ($\omega, \alpha$), invariance, attractors, basins of attraction.
    \item Periodic orbits & limit cycles. Poincaré-Bendixson (2D). Poincaré sections (higher D).
    \item Bifurcations: Qualitative changes as parameters vary (Saddle-Node, Transcritical, Pitchfork, Hopf, Period-Doubling).
    \item Chaos: Sensitive dependence, positive Lyapunov exponents, strange attractors, fractal dimension.
    \end{itemize}
    \item Computational tools (simulation, data analysis) are crucial.
    \item Rich applications across science and engineering.
  \end{itemize}
\end{frame}

\begin{frame}{Further Topics / Research Directions}
  \frametitle{Beyond the Basics}

  \begin{itemize}
    \item Hamiltonian Dynamics (conservation of volume, KAM theory) vs Dissipative Dynamics.
    \item Ergodic Theory (statistical properties of dynamics).
    \item Control Theory (influencing dynamics).
    \item Synchronization Phenomena.
    \item Dynamics on Networks or Graphs.
    \item Spatiotemporal Dynamics (PDEs, Lattice models, Cellular Automata).
    \item Random Dynamical Systems / Stochastic Systems.
    \item Delay Differential Equations.
    \item Data-Driven Discovery of Dynamics.
  \end{itemize}

  % \note{Speaker notes: These are areas where students might find thesis topics or see connections to their specific field.}
\end{frame}

\begin{frame}
  \frametitle{Thank You}
  \centering
  \Large Thank you for your attention!\\
  Questions?

  \vspace{1cm}
  {\small These slides provide an overview. Delving into proofs, detailed examples, and specific computational methods will be covered in subsequent lectures.}
\end{frame}


%--------------------
% END OF DOCUMENT
%--------------------
\end{document}
```

**To Use These Slides:**

1.  **Save:** Save the code as a `.tex` file (e.g., `dynamical_systems_course.tex`).
2.  **Figures:** Create a subdirectory named `figures` in the same directory as your `.tex` file. You will need to generate or find images for the placeholders (e.g., phase portraits, bifurcation diagrams, attractors). Replace the placeholder text like `Figure Placeholder: Stable Node Phase Portrait` with actual `\includegraphics` commands pointing to your image files (e.g., `\includegraphics[width=\textwidth]{node.pdf}`). I recommend generating phase portraits and bifurcation diagrams using software like MATLAB, Python (with matplotlib/SciPy), or specialized dynamical systems tools.
3.  **Compile:** Compile the `.tex` file using a LaTeX compiler (like `pdflatex`). You might need to compile twice for the table of contents or cross-references to be correct.
    ```bash
    pdflatex dynamical_systems_course.tex
    pdflatex dynamical_systems_course.tex
    ```
4.  **Customize:**
    *   Change `Your Name`, `Your Department/Institution` on the title page.
    *   Uncomment and fill in the `\note{}` sections for speaker notes.
    *   Adjust the theme and colors (`\usetheme`, `\usecolortheme`).
    *   Add or remove specific examples relevant to the class's focus or the students' background.
    *   Fill in details (equations, specific values, proofs) where summaries are currently provided.
    *   Consider adding `\pause` commands within `itemize` or `enumerate` environments to reveal items one by one during the presentation.
    *   Add actual problem examples or derivations as separate slides or within existing ones.

This structure provides a solid foundation covering the breadth of a graduate-level introductory course on dynamical systems theory, tailored for a diverse technical audience. Good luck with your lectures!

Of course. Here is a comprehensive set of lecture slides in LaTeX Beamer format for a graduate-level course on Dynamical Systems Theory. The slides are structured as a series of five introductory lectures, covering the core concepts from fundamentals to chaos and applications.

The content is tailored for a PhD-level audience from Physics, Applied Math, Computer Science, and Electrical & Computer Engineering (ECE). It balances mathematical rigor with physical intuition and highlights applications relevant to each field.

### Key Features of these Slides:

*   **Modular Structure:** The material is broken down into five logical lectures, each in its own `.tex` file for easy management.
*   **Modern Beamer Theme:** Uses the `Madrid` theme, which is clean and professional with a useful navigation sidebar.
*   **Audience-Specific Content:** Examples are drawn from various disciplines (pendulum, circuits, population dynamics, neural networks).
*   **Placeholders for Figures:** The code includes placeholders (`\includegraphics[...]{...}`) and detailed comments on what kind of figure should be placed there. You will need to generate these figures using tools like Python (with Matplotlib/SciPy), MATLAB, or similar.
*   **Presenter Notes:** The `\note{...}` command is used to add notes for the lecturer, which are not visible on the slides themselves.
*   **Comprehensive Coverage:** The lectures progress from basic definitions to fixed-point analysis, bifurcations, phase portraits, limit cycles, and finally to chaos and strange attractors.

---

### How to Use This Material

1.  **Create a Main File (`main.tex`):** This file will load the Beamer class and include the individual lecture files.
2.  **Create Lecture Files:** Create separate `.tex` files for each lecture (`lecture1.tex`, `lecture2.tex`, etc.).
3.  **Create a Figures Folder:** Make a folder named `figures` in the same directory. You will need to create and place your `.png` or `.pdf` plots in this folder.
4.  **Compile:** Compile `main.tex` with a LaTeX distribution that has Beamer installed (like TeX Live, MiKTeX). You will likely need to compile it twice to get the table of contents correct.

---

### File 1: `main.tex`

```latex
\documentclass{beamer}

%------------------------------------------------------------
% Preamble
%------------------------------------------------------------
\usepackage[utf8]{inputenc}
\usepackage{amsmath, amssymb, amsfonts} % For advanced math
\usepackage{graphicx}       % To include images
\usepackage{bm}             % For bold math symbols (\bm)
\usepackage{pgfplots}       % For drawing plots directly in LaTeX (optional)
\pgfplotsset{compat=1.17}

% Beamer Theme
\usetheme{Madrid}
\usecolortheme{default} % or dolphin, whale, seahorse

% Information for the title page
\title[Dynamical Systems]{An Introduction to Dynamical Systems Theory}
\author{Your Name}
\institute{Department of XYZ \\ University of ABC}
\date{\today}

%------------------------------------------------------------
% Document
%------------------------------------------------------------
\begin{document}

% Title Page
\begin{frame}
    \titlepage
\end{frame}

% Table of Contents
\begin{frame}{Outline of the Course}
    \tableofcontents
\end{frame}

% Include the lecture files
\input{lecture1.tex}
\input{lecture2.tex}
\input{lecture3.tex}
\input{lecture4.tex}
\input{lecture5.tex}

\end{document}
```

---

### File 2: `lecture1.tex` (Lecture 1: Introduction and Fundamentals)

```latex
% lecture1.tex
\section{Lecture 1: Introduction and Fundamentals}

\begin{frame}{What is a Dynamical System?}
    \begin{block}{Definition}
        A dynamical system consists of two parts:
        \begin{enumerate}
            \item \textbf{A state space} $\mathcal{S}$: The set of all possible states of the system. A state is a vector $\bm{x} \in \mathcal{S} \subseteq \mathbb{R}^n$.
            \item \textbf{An evolution rule} $\phi$: A function that describes how the state evolves over time.
        \end{enumerate}
    \end{block}

    \vfill

    We primarily study two types:
    \begin{itemize}
        \item \textbf{Continuous-Time Systems (Flows):} Described by ordinary differential equations (ODEs).
        \[ \dot{\bm{x}} = \frac{d\bm{x}}{dt} = f(\bm{x}, t), \quad \bm{x} \in \mathbb{R}^n \]
        The evolution rule is the solution $\phi_t(\bm{x}_0)$ that gives the state at time $t$.

        \item \textbf{Discrete-Time Systems (Maps):} Described by difference equations or iterated maps.
        \[ \bm{x}_{k+1} = f(\bm{x}_k), \quad \bm{x} \in \mathbb{R}^n \]
        The evolution rule is the repeated application of the map $f$.
    \end{itemize}
    \note{Emphasize that the state vector x contains all information needed to uniquely determine the future state.}
\end{frame}

\begin{frame}{Why Study Dynamical Systems? Applications}
    Dynamical systems provide a universal language for modeling time-dependent phenomena.

    \begin{columns}
        \begin{column}{0.5\textwidth}
            \textbf{Physics:}
            \begin{itemize}
                \item Celestial mechanics (planetary orbits)
                \item Fluid dynamics (weather prediction)
                \item Classical mechanics (pendulum)
                \item Plasma physics
            \end{itemize}
            \textbf{ECE:}
            \begin{itemize}
                \item Electronic circuits (oscillators)
                \item Control theory (robotics, autopilots)
                \item Signal processing
                \item Communication networks
            \end{itemize}
        \end{column}
        \begin{column}{0.5\textwidth}
            \textbf{Applied Math:}
            \begin{itemize}
                \item Population dynamics (predator-prey)
                \item Epidemiology (disease spread)
                \item Stability theory
            \end{itemize}
            \textbf{Computer Science:}
            \begin{itemize}
                \item Neural networks (learning dynamics)
                \item Algorithm analysis (e.g., gradient descent)
                \item Network science (information cascades)
            \end{itemize}
        \end{column}
    \end{columns}
\end{frame}

\begin{frame}{Key Concepts: Phase Space and Trajectories}
    \begin{columns}
        \begin{column}{0.5\textwidth}
            \textbf{Phase Space (or State Space):}
            \begin{itemize}
                \item An abstract space where each coordinate represents a variable needed to describe the state.
                \item For a simple pendulum, the state is $(\theta, \dot{\theta})$. The phase space is a 2D cylinder.
                \item At any instant, the system's state is a single point in this space.
            \end{itemize}

            \textbf{Trajectory (or Orbit):}
            \begin{itemize}
                \item The path traced by the state point as it evolves in time.
                \item A solution $\bm{x}(t)$ to an ODE starting from $\bm{x}_0$ forms a trajectory.
                \item For deterministic systems, trajectories cannot cross. Why? (Uniqueness of solutions).
            \end{itemize}
        \end{column}
        \begin{column}{0.5\textwidth}
            \centering
            % --- FIGURE ---
            % Description: A diagram of a 2D phase space (e.g., x-y plane).
            % Show a few initial points (x0, y0) and the trajectories (curves with arrows)
            % emanating from them, illustrating the concept of a flow.
            \includegraphics[width=\textwidth]{figures/phase_space_example.png}
            \caption{A phase portrait showing trajectories in a 2D phase space.}
        \end{column}
    \end{columns}
\end{frame}

\begin{frame}{Canonical Examples}
    \begin{block}{The Simple Pendulum (Physics, ECE)}
        Equation of motion: $m L \ddot{\theta} + b \dot{\theta} + m g \sin\theta = 0$.
        \vfill
        Let $x = \theta$ and $y = \dot{\theta}$. The system becomes:
        \[
        \begin{aligned}
        \dot{x} &= y \\
        \dot{y} &= -\frac{g}{L}\sin x - \frac{b}{mL}y
        \end{aligned}
        \]
        This is a 2D continuous-time system.
    \end{block}
    \vfill
    \begin{block}{The Logistic Map (Biology, CS)}
        A simple model for population growth with limited resources:
        \[ x_{k+1} = r x_k (1 - x_k) \]
        \begin{itemize}
            \item $x_k \in [0, 1]$ is the population at generation $k$, normalized.
            \item $r > 0$ is a parameter representing the growth rate.
        \end{itemize}
        This is a 1D discrete-time system that exhibits surprisingly complex behavior.
    \end{block}
\end{frame}
```

---

### File 3: `lecture2.tex` (Lecture 2: 1D Systems and Bifurcations)

```latex
% lecture2.tex
\section{Lecture 2: 1D Systems and Bifurcations}

\begin{frame}{One-Dimensional Flows: $\dot{x} = f(x)$}
    The simplest continuous systems. Dynamics are limited, but crucial for building intuition.
    \begin{itemize}
        \item Motion is restricted to a line.
        \item The state $x(t)$ can only move left ($f(x)<0$) or right ($f(x)>0$).
        \item Trajectories can't oscillate. They must either approach a constant value or go to $\pm\infty$.
    \end{itemize}

    \begin{block}{Fixed Points (Equilibria)}
        A point $x^*$ is a \textbf{fixed point} if the system stays there forever once it arrives.
        \[ \dot{x} = f(x^*) = 0 \]
    \end{block}
    Geometrically, fixed points are the zeros of the function $f(x)$.

    \centering
    % --- FIGURE ---
    % Description: A plot of a function f(x) vs x. Mark the points where f(x)=0 as x^*.
    % On the x-axis below, draw arrows indicating the direction of flow (right if f(x)>0, left if f(x)<0).
    \includegraphics[width=0.7\textwidth]{figures/flow_on_a_line.png}
\end{frame}

\begin{frame}{Stability of Fixed Points}
    What happens if we perturb the system slightly from a fixed point $x^*$?

    \begin{itemize}
        \item \textbf{Stable}: If the system returns to $x^*$.
        \item \textbf{Unstable}: If the system moves away from $x^*$.
        \item \textbf{Semi-stable}: If stable on one side and unstable on the other.
    \end{itemize}

    \begin{block}{Linear Stability Analysis}
        Consider a small perturbation $\eta(t) = x(t) - x^*$.
        \[ \dot{\eta} = \dot{x} = f(x^* + \eta) \approx f(x^*) + f'(x^*) \eta = f'(x^*) \eta \]
        The solution is $\eta(t) \approx \eta(0) e^{f'(x^*)t}$.
    \end{block}

    \begin{itemize}
        \item If $f'(x^*) < 0$, the perturbation decays $\implies$ \textbf{stable}.
        \item If $f'(x^*) > 0$, the perturbation grows $\implies$ \textbf{unstable}.
        \item If $f'(x^*) = 0$, the linear analysis is inconclusive (need to check higher-order terms).
    \end{itemize}
    \note{This is a cornerstone of stability theory, used in nearly every field. Mention linearization is a local approximation.}
\end{frame}

\begin{frame}{Bifurcations: The Birth of New Dynamics}
    \begin{block}{Definition}
        A \textbf{bifurcation} is a qualitative change in the long-term behavior of a system as a parameter is varied. Typically, this involves the creation or destruction of fixed points, or a change in their stability.
    \end{block}

    \begin{columns}
    \begin{column}{0.5\textwidth}
        \textbf{Saddle-Node Bifurcation}\\
        Normal Form: $\dot{x} = r + x^2$.
        \begin{itemize}
            \item $r<0$: Two fixed points (stable, unstable).
            \item $r=0$: One semi-stable fixed point.
            \item $r>0$: No fixed points.
        \end{itemize}
    \end{column}
    \begin{column}{0.5\textwidth}
        \centering
        % --- FIGURE ---
        % Description: A bifurcation diagram for the saddle-node.
        % Plot x^* vs r. It should look like a parabola opening to the left.
        % Use solid lines for stable branches and dashed lines for unstable branches.
        \includegraphics[width=\textwidth]{figures/saddle_node_bifurcation.png}
    \end{column}
    \end{columns}
    \note{The normal form is the simplest possible equation that captures the essence of the bifurcation.}
\end{frame}

\begin{frame}{Other Common 1D Bifurcations}
    \begin{columns}
        \begin{column}{0.5\textwidth}
            \textbf{Transcritical Bifurcation}\\
            Normal Form: $\dot{x} = rx - x^2$.
            \begin{itemize}
                \item Fixed points exchange stability.
                \item Common in population models.
            \end{itemize}
             \centering
             % --- FIGURE ---
             % Description: Bifurcation diagram for transcritical.
             % Two lines, x^*=0 and x^*=r, intersecting at r=0.
             % Show the stability of each branch changing as r passes through 0.
             \includegraphics[width=\textwidth]{figures/transcritical_bifurcation.png}
        \end{column}
        \begin{column}{0.5\textwidth}
            \textbf{Pitchfork Bifurcation (Supercritical)}\\
            Normal Form: $\dot{x} = rx - x^3$.
            \begin{itemize}
                \item One stable fixed point gives way to an unstable one and two new stable ones.
                \item A model for phase transitions (e.g., magnetization).
            \end{itemize}
            \centering
            % --- FIGURE ---
            % Description: Bifurcation diagram for supercritical pitchfork.
            % A horizontal line for x^*=0. At r=0, it becomes unstable (dashed)
            % and a stable parabola x^* = +/-sqrt(r) emerges for r>0.
            \includegraphics[width=\textwidth]{figures/pitchfork_bifurcation.png}
        \end{column}
    \end{columns}
\end{frame}

\begin{frame}{The Logistic Map: A Route to Chaos}
    Back to our discrete map: $x_{k+1} = r x_k (1-x_k)$.

    As we increase the parameter $r$, the long-term behavior becomes increasingly complex.
    \begin{itemize}
        \item For small $r$, the population converges to a single fixed point.
        \item At $r=3$, the fixed point becomes unstable and a stable 2-cycle appears (population alternates between two values).
        \item This is followed by a 4-cycle, 8-cycle, etc. This is called a \textbf{period-doubling cascade}.
        \item At $r_{\infty} \approx 3.56995...$, the system becomes \textbf{chaotic}.
    \end{itemize}

    \centering
    % --- FIGURE ---
    % Description: The famous bifurcation diagram for the logistic map.
    % Plot the long-term values of x_k (the attractor) versus the parameter r.
    % This will show the fixed point, the period-doubling cascade, and regions of chaos.
    \includegraphics[width=0.9\textwidth]{figures/logistic_map_bifurcation.png}
    \caption{The Feigenbaum Diagram: Universal route to chaos.}
\end{frame}
```

---

### File 4: `lecture3.tex` (Lecture 3: 2D Systems and Limit Cycles)

```latex
% lecture3.tex
\section{Lecture 3: 2D Systems and Limit Cycles}

\begin{frame}{Two-Dimensional Flows: $\dot{\bm{x}} = f(\bm{x})$}
    In 2D, $\bm{x}=(x,y)$, we can have much richer dynamics than in 1D.
    \[
    \begin{aligned}
    \dot{x} &= f(x,y) \\
    \dot{y} &= g(x,y)
    \end{aligned}
    \]
    \begin{itemize}
        \item Fixed points still exist where $f(x^*, y^*) = 0$ and $g(x^*, y^*) = 0$.
        \item Trajectories can now spiral, rotate, and form closed loops.
        \item The \textbf{phase portrait} (a drawing of trajectories in the phase plane) is our main tool.
    \end{itemize}
    
    \begin{block}{Linearization around a Fixed Point}
        Let $\bm{x}^*$ be a fixed point. For a small perturbation $\bm{\eta} = \bm{x} - \bm{x}^*$:
        \[ \dot{\bm{\eta}} \approx J \bm{\eta}, \quad \text{where } J = \begin{pmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{pmatrix}_{\bm{x}=\bm{x}^*} \]
        $J$ is the \textbf{Jacobian matrix}. The local behavior is determined by the eigenvalues ($\lambda_1, \lambda_2$) of $J$.
    \end{block}
\end{frame}

\begin{frame}{Classification of Linear Systems}
    The behavior of $\dot{\bm{x}} = A\bm{x}$ depends on the eigenvalues of $A$. Let $\tau = \text{tr}(A)$ and $\Delta = \det(A)$. The eigenvalues are $\lambda = \frac{\tau \pm \sqrt{\tau^2 - 4\Delta}}{2}$.

    \begin{itemize}
    \item \textbf{Saddles}: $\Delta < 0$. Eigenvalues are real with opposite signs ($\lambda_1 < 0 < \lambda_2$). Always unstable.
    \item \textbf{Nodes}: $\Delta > 0$ and $\tau^2 \ge 4\Delta$. Eigenvalues are real with the same sign. Stable if $\tau < 0$, unstable if $\tau > 0$.
    \item \textbf{Spirals (Foci)}: $\Delta > 0$ and $\tau^2 < 4\Delta$. Eigenvalues are complex conjugates. Stable if $\tau < 0$, unstable if $\tau > 0$.
    \item \textbf{Centers}: $\tau = 0$ and $\Delta > 0$. Eigenvalues are purely imaginary. Neutrally stable.
    \end{itemize}
    
    \centering
    % --- FIGURE ---
    % Description: The famous Trace-Determinant plane. A 2D plot with tau on the x-axis
    % and Delta on the y-axis. The parabola tau^2 - 4*Delta = 0 separates nodes from spirals.
    % The vertical axis (tau=0) separates stable from unstable. The horizontal axis (Delta=0)
    % separates saddles from everything else. Label the regions (Stable Node, Unstable Spiral, etc.).
    \includegraphics[width=0.8\textwidth]{figures/trace_determinant_plane.png}
\end{frame}

\begin{frame}{Nonlinear Systems: Predator-Prey Model}
    \begin{block}{The Lotka-Volterra Equations}
        Model for rabbits ($x$) and foxes ($y$):
        \[
        \begin{aligned}
        \dot{x} &= ax - bxy & (\text{Rabbits grow, get eaten by foxes}) \\
        \dot{y} &= -cy + dxy & (\text{Foxes die, grow by eating rabbits})
        \end{aligned}
        \]
        where $a,b,c,d > 0$.
    \end{block}

    \textbf{Analysis:}
    \begin{itemize}
        \item \textbf{Nullclines}: $\dot{x}=0 \implies x=0$ or $y=a/b$. $\dot{y}=0 \implies y=0$ or $x=c/d$.
        \item \textbf{Fixed Points}:
        \begin{enumerate}
            \item $(0,0)$: Trivial extinction. (Saddle point)
            \item $(c/d, a/b)$: Coexistence. (Center in the linear approximation)
        \end{enumerate}
    \end{itemize}
    The full nonlinear system shows persistent, nested oscillations (closed orbits) around the coexistence point.
    
    \centering
    % --- FIGURE ---
    % Description: Phase portrait for the Lotka-Volterra system. Show the two fixed points.
    % Show the closed, periodic orbits spiraling around the center point (c/d, a/b).
    % Also include arrows on the axes showing the flow.
    \includegraphics[width=0.6\textwidth]{figures/lotka_volterra.png}
\end{frame}

\begin{frame}{Limit Cycles}
    \begin{block}{Definition}
    A \textbf{limit cycle} is an isolated closed trajectory. "Isolated" means that neighboring trajectories are not closed; they spiral towards or away from the limit cycle.
    \end{block}
    
    \begin{itemize}
        \item They represent sustained, stable oscillations.
        \item Cannot occur in 1D systems.
        \item Unlike the Lotka-Volterra centers, the amplitude and frequency of oscillation are determined by the system itself, not the initial conditions.
        \item Crucial in electronics (oscillators), biology (circadian rhythms), and chemistry (chemical clocks).
    \end{itemize}

    \begin{alertblock}{The Poincaré-Bendixson Theorem}
        A powerful result for 2D systems. If a trajectory is trapped in a closed, bounded region $\mathcal{R}$ that contains no fixed points, then the trajectory must approach a closed orbit (a limit cycle).
    \end{alertblock}
    \note{Mention the limitations: it does not apply to 3D or higher, which is why chaos is possible in 3D but not 2D flows.}
\end{frame}
```

---

### File 5: `lecture4.tex` (Lecture 4: Chaos and Strange Attractors)

```latex
% lecture4.tex
\section{Lecture 4: Chaos and Strange Attractors}

\begin{frame}{What is Chaos?}
    Chaos is aperiodic, long-term, bounded behavior in a deterministic system that exhibits sensitive dependence on initial conditions.

    \begin{block}{The Three Ingredients of Chaos}
    \begin{enumerate}
        \item \textbf{Sensitive Dependence on Initial Conditions (SDIC):}
        Nearby initial conditions diverge exponentially, on average. The "Butterfly Effect". This makes long-term prediction impossible.
        
        \item \textbf{Topological Transitivity (Mixing):}
        The system will eventually explore all regions of its phase space. Trajectories are not "stuck" in one area.
        
        \item \textbf{Dense Periodic Orbits:}
        Embedded within the chaotic region is an infinite number of unstable periodic orbits. Chaos is "close" to being periodic, but never settles.
    \end{enumerate}
    \end{block}
    \note{Emphasize that chaos is NOT random. It is generated by a deterministic rule.}
\end{frame}

\begin{frame}{Quantifying Chaos: Lyapunov Exponents}
    SDIC can be quantified by the \textbf{Lyapunov exponent}, $\lambda$.
    Consider two nearby initial conditions $\bm{x}_0$ and $\bm{x}_0 + \bm{\delta}_0$. The separation vector $\bm{\delta}(t)$ between their trajectories evolves approximately as:
    \[ \|\bm{\delta}(t)\| \approx \|\bm{\delta}_0\| e^{\lambda t} \]

    \begin{itemize}
        \item $\lambda > 0$: Exponential separation. \textbf{This is the signature of chaos}.
        \item $\lambda = 0$: Linear separation (e.g., neutrally stable orbit).
        \item $\lambda < 0$: Exponential convergence (e.g., stable fixed point or limit cycle).
    \end{itemize}

    For an $n$-dimensional system, there is a whole \textbf{spectrum} of Lyapunov exponents $(\lambda_1, \lambda_2, ..., \lambda_n)$ corresponding to different directions of perturbation.
    The presence of at least one positive Lyapunov exponent is the standard diagnostic for chaos.
\end{frame}

\begin{frame}{The Lorenz System: A Paradigm of Chaos}
    A simplified model of atmospheric convection, introduced by Edward Lorenz in 1963.
    \begin{block}{The Lorenz Equations}
        \[
        \begin{aligned}
        \dot{x} &= \sigma(y - x) \\
        \dot{y} &= x(\rho - z) - y \\
        \dot{z} &= xy - \beta z
        \end{aligned}
        \]
        Standard chaotic parameters: $\sigma=10, \rho=28, \beta=8/3$.
    \end{block}

    \begin{columns}
        \begin{column}{0.5\textwidth}
            \begin{itemize}
                \item A 3D system, the minimum dimension for chaos in flows.
                \item Has three fixed points, all of which are unstable.
                \item Trajectories are bounded but never repeat and never settle down.
                \item Lyapunov spectrum: $(\lambda_1, \lambda_2, \lambda_3) \approx (0.9, 0, -14.6)$. The positive $\lambda_1$ confirms chaos.
            \end{itemize}
        \end{column}
        \begin{column}{0.5\textwidth}
            \centering
            % --- FIGURE ---
            % Description: The iconic "butterfly" attractor of the Lorenz system.
            % A 3D plot showing a long trajectory. The two "wings" correspond to
            % clockwise and counter-clockwise convection rolls.
            \includegraphics[width=\textwidth]{figures/lorenz_attractor.png}
            \caption{The Lorenz Attractor}
        \end{column}
    \end{columns}
\end{frame}

\begin{frame}{Strange Attractors}
    \begin{block}{Definition}
    An attractor that exhibits chaotic dynamics. It has a geometric structure that is \textbf{fractal}.
    \end{block}
    
    \textbf{Properties:}
    \begin{itemize}
        \item \textbf{Attractor:} It is an attracting set; nearby trajectories converge to it.
        \item \textbf{Strange:}
        \begin{enumerate}
            \item \textit{Dynamics are chaotic:} It exhibits SDIC ($\lambda > 0$).
            \item \textit{Geometry is fractal:} It has a non-integer (fractal) dimension. It contains structure at arbitrarily small scales.
        \end{enumerate}
    \end{itemize}

    \begin{alertblock}{The Geometry of Chaos}
    The fractal nature is a consequence of the dynamics. To allow for infinite stretching (for SDIC) within a finite, bounded volume, the attractor must repeatedly be stretched and folded.
    \end{alertblock}

    \centering
    % --- FIGURE ---
    % Description: A diagram illustrating the stretch-and-fold mechanism.
    % Show a small volume of initial conditions in phase space.
    % Step 1: Stretch it in one direction (due to positive Lyapunov exp).
    % Step 2: Squeeze it in another direction (to stay bounded).
    % Step 3: Fold it back over on itself. Repeat.
    \includegraphics[width=0.7\textwidth]{figures/stretch_and_fold.png}
\end{frame}
```
---

### File 6: `lecture5.tex` (Lecture 5: Advanced Topics and Applications)

```latex
% lecture5.tex
\section{Lecture 5: Advanced Topics \& Applications}

\begin{frame}{Hamiltonian Systems (Physics)}
    Describes systems that conserve energy (e.g., lossless pendulum, planetary motion).
    
    \begin{itemize}
        \item Defined by a Hamiltonian function $H(\bm{q}, \bm{p})$, where $\bm{q}$ are positions and $\bm{p}$ are momenta.
        \item Equations of motion: $\dot{q}_i = \frac{\partial H}{\partial p_i}$, $\dot{p}_i = -\frac{\partial H}{\partial q_i}$.
        \item \textbf{Liouville's Theorem}: Phase space volume is conserved. The flow is "incompressible".
        \item \textbf{No attractors!}: Since volume is conserved, trajectories cannot converge onto a smaller-dimensional set. No fixed point attractors, no limit cycles, no strange attractors.
        \item Dynamics are typically quasiperiodic (on tori) or chaotic.
    \end{itemize}
    
    \begin{block}{KAM Theorem (Kolmogorov-Arnold-Moser)}
        A profound result stating that many quasiperiodic orbits of an integrable Hamiltonian system persist under small, smooth perturbations. This explains the stability of systems like our solar system.
    \end{block}
\end{frame}

\begin{frame}{Synchronization and Control (ECE, CS)}
    Once we understand chaos, can we use or control it?
    
    \begin{block}{Control of Chaos}
    Goal: Use tiny, carefully chosen perturbations to stabilize one of the infinite unstable periodic orbits (UPOs) embedded within a strange attractor.
    \begin{itemize}
        \item \textbf{OGY Method (Ott, Grebogi, Yorke):} The seminal idea.
        \item Wait for the chaotic trajectory to come near the desired UPO.
        \item Apply a small kick to a system parameter to nudge it onto the UPO's stable manifold.
        \item Application: Stabilizing lasers, controlling chemical reactions, stabilizing cardiac arrhythmias (in theory).
    \end{itemize}
    \end{block}
    
    \begin{block}{Synchronization}
    When two or more chaotic systems are coupled, they can sometimes synchronize their behavior.
    \begin{itemize}
        \item \textbf{Identical Synchronization:} $\bm{x}_1(t) = \bm{x}_2(t)$.
        \item Key for secure communications: use one chaotic circuit as a transmitter and a synchronized circuit as a receiver to decode a message masked by chaos.
        \item \textbf{Kuramoto Model:} A model for synchronization in large populations of coupled oscillators, relevant to neuroscience (brain waves) and power grids.
    \end{itemize}
    \end{block}
\end{frame}

\begin{frame}{Network Dynamics (CS, Applied Math)}
    Applying dynamical systems thinking to the structure and function of networks.

    \begin{columns}
    \begin{column}{0.5\textwidth}
        \textbf{Dynamics \textit{on} Networks}
        \begin{itemize}
            \item The network topology is fixed.
            \item Nodes have states that evolve based on their neighbors.
            \item \textbf{Examples:}
            \begin{itemize}
                \item SIS/SIR models of epidemic spread on a social network.
                \item Consensus algorithms in distributed computing.
                \item Opinion dynamics.
            \end{itemize}
            \item Questions: How does network structure (e.g., hubs) affect stability, spreading speed, etc.?
        \end{itemize}
    \end{column}
    \begin{column}{0.5\textwidth}
        \textbf{Dynamics \textit{of} Networks}
        \begin{itemize}
            \item The network structure itself evolves.
            \item Edges are created or destroyed based on node states or other rules.
            \item \textbf{Examples:}
            \begin{itemize}
                \item Hebbian learning in neural networks ("neurons that fire together, wire together").
                \item Adaptive social networks.
            \end{itemize}
            \item Co-evolution of structure and dynamics.
        \end{itemize}
    \end{column}
    \end{columns}
\end{frame}

\begin{frame}{Course Summary and Outlook}
    \textbf{What We've Covered:}
    \begin{itemize}
        \item The fundamental framework of maps and flows.
        \item The crucial role of fixed points and their stability.
        \item Bifurcations as the source of qualitative change.
        \item Rich behaviors in 2D: phase portraits and limit cycles.
        \item The discovery and characterization of chaos and strange attractors.
        \item Glimpses into advanced applications across disciplines.
    \end{itemize}
    
    \vfill
    
    \textbf{Where to Go From Here:}
    \begin{itemize}
        \item \textbf{Infinite-Dimensional Systems (PDEs):} Spatiotemporal chaos, patterns (e.g., Turing patterns).
        \item \textbf{Stochastic Dynamical Systems:} The role of noise.
        \item \textbf{Ergodic Theory:} The mathematical foundation connecting dynamics to statistics and measure theory.
        \item \textbf{Data-Driven Discovery:} Using modern machine learning to discover dynamical systems from time-series data (e.g., SINDy).
    \end{itemize}
\end{frame}

\begin{frame}
    \centering
    \Huge Thank You \par
    \Large Questions?
\end{frame}
```

Absolutely! Below is a **comprehensive LaTeX Beamer template** for a graduate-level course on Dynamical Systems Theory, targeting an advanced, interdisciplinary audience. This slide deck covers foundational concepts, key theorems, examples, and pointers for further study.

You can **copy & save this as e.g., `dynamical_systems_lecture1.tex`**. Compile it with `pdflatex` or an equivalent LaTeX compiler that supports Beamer.

---

```latex
\documentclass[aspectratio=169]{beamer}
\usetheme{Boadilla}
\usecolortheme{dolphin}
\usepackage{amsmath, amssymb, amsfonts, physics, graphicx}
\usepackage{hyperref}
\usepackage{tikz}

\title{Introduction to Dynamical Systems Theory}
\subtitle{Graduate Lecture Series}
\author{Your Name}
\institute{Department of Physics / Mathematics / ECE / Computer Science}
\date{\today}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \titlepage
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Course Overview}
  \tableofcontents
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Motivation and Examples}
\begin{frame}{What are Dynamical Systems?}
\begin{itemize}
    \item \textbf{Dynamical System:} A mathematical model describing the evolution of a state over time.
    \item \textbf{State Space:} The set of all possible states of a system.
    \item \textbf{Classification:}
    \begin{itemize}
        \item Discrete-time vs. continuous-time
        \item Finite-dimensional vs. infinite-dimensional
        \item Deterministic vs. stochastic
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Canonical Examples}
\begin{itemize}
    \item Population dynamics (\textbf{Logistic map, Lotka-Volterra})
    \item Classical mechanics (Pendulum, double pendulum)
    \item Electrical circuits (\textbf{RLC circuit})
    \item Neural models (\textbf{Hodgkin-Huxley, Izhikevich})
    \item Weather/Climate (\textbf{Lorenz system})
    \item Control systems, robotics
\end{itemize}
\pause
\begin{block}{Key Reference}
    S.H. Strogatz, \emph{Nonlinear Dynamics and Chaos} (2018)
\end{block}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Types of Dynamical Systems}

\begin{frame}{Mathematical Formulation}
\textbf{Continuous-time:}
\[
\frac{d\vec{x}}{dt} = \vec{F}(\vec{x}, t), \qquad \vec{x}(0) = \vec{x}_0
\]
\textbf{Discrete-time:}
\[
\vec{x}_{n+1} = G(\vec{x}_n), \qquad \vec{x}_0 \text{ given}
\]
\pause
\begin{block}{State Space $\mathcal{M}$}
$\vec{x}(t) \in \mathcal{M}$, where $\mathcal{M}$ is typically $\mathbb{R}^n$ or a manifold.
\end{block}
\end{frame}

\begin{frame}{Flows and Maps}
\begin{itemize}
    \item \textbf{Flow:} A family $\Phi^t$ of transformations: $\Phi^t(\vec{x}_0)$ solves the ODE for any $t$.
    \item \textbf{Map:} Discrete updates; iteration of $G$.
    \item \textbf{Autonomous} ($\vec{F}$ independent of $t$), \textbf{Non-autonomous}
    \item \textbf{Determinism:} The future is uniquely determined by current state.
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Qualitative Theory: Fixed Points and Stability}
\begin{frame}{Fixed Points and Periodic Orbits}
\begin{block}{Definitions}
\begin{itemize}
    \item \textbf{Fixed Point} (Equilibrium): $\vec{F}(\vec{x}^*) = 0$
    \item \textbf{Periodic Orbit:} There exists $T>0$ s.t. $\Phi^T(\vec{x}_0) = \vec{x}_0$
\end{itemize}
\end{block}
\pause
\textbf{Stability:} Small perturbations decay, grow, or remain neutral.
\end{frame}

\begin{frame}{Linear Stability Analysis}
\begin{itemize}
    \item Linearize around fixed point $\vec{x}^*$:
    \[
    \frac{d\vec{\xi}}{dt} = D\vec{F}(\vec{x}^*) \vec{\xi}, \; \vec{\xi} = \vec{x} - \vec{x}^*
    \]
    \item $D\vec{F}$ is Jacobian matrix.
    \item Solution: $\vec{\xi}(t) = \exp(D\vec{F} \, t)\vec{\xi}(0)$
    \item Stability determined by eigenvalues $\lambda_i$ of $D\vec{F}$:
      \begin{itemize}
        \item $\Re(\lambda_i) < 0$ for all $i$ $\Rightarrow$ stable
        \item $\Re(\lambda_j) > 0$ for some $j$ $\Rightarrow$ unstable
      \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Example: The Logistic Map}
\[
x_{n+1} = r x_n (1-x_n)
\]
\begin{itemize}
    \item Discrete-time nonlinear map, modeling population growth.
    \item Vary $r$ to see transitions: fixed points, periodic points, chaos.
\end{itemize}
\includegraphics[width=0.7\linewidth]{logmap_bifurcation.png}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Topological and Geometrical Aspects}

\begin{frame}{Phase Portraits}
\begin{itemize}
    \item Geometric representation of trajectories in phase space.
    \item Nullclines, vector fields, flow lines
\end{itemize}
\begin{center}
\begin{tikzpicture}[scale=1]
\draw[->] (-2,0) -- (2,0) node[right] {$x$};
\draw[->] (0,-2) -- (0,2) node[above] {$y$};
\foreach \x in {-1,0,1}
    \foreach \y in {-1,0,1}
        \draw[->,thick] (\x,\y) -- ({\x+0.15*\x-0.15*\y},{\y+0.1*\x+0.13*\y});
\end{tikzpicture}
\end{center}
\end{frame}

\begin{frame}{Limit Cycles and Attractors}
\begin{itemize}
    \item \textbf{Limit Cycle:} Isolated closed trajectory
    \item \textbf{Attractor:} Set towards which system evolves
    \item \textbf{Types:} Fixed point, limit cycle, torus, strange (chaotic) attractor
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Bifurcation Theory}

\begin{frame}{Parameter Dependence and Bifurcations}
\begin{itemize}
    \item \textbf{Bifurcation:} Qualitative change of dynamics as a parameter is varied.
    \item \textbf{Types:}
    \begin{itemize}
        \item Saddle-node (fold)
        \item Pitchfork (super/subcritical)
        \item Hopf (oscillatory instability)
        \item Period-doubling
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Example: Saddle-Node Bifurcation}
\[
\dot{x} = \mu - x^2
\]
\begin{itemize}
    \item For $\mu < 0$: no real fixed point
    \item For $\mu > 0$: two fixed points ($\pm\sqrt{\mu}$), stability changes
\end{itemize}
\begin{center}
\includegraphics[width=0.45\linewidth]{saddlenode_bifurcation.png}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Chaos, Sensitivity, and Strange Attractors}

\begin{frame}{What is Chaos?}
\begin{itemize}
    \item Extreme sensitivity to initial conditions
    \item Determinism without predictability
    \item Dense periodic orbits, topological transitivity
\end{itemize}
\pause
\textbf{Example:} The Lorenz system
\[
\dot{x} = \sigma(y-x), \;\; \dot{y} = x(\rho-z)-y, \;\; \dot{z} = xy-\beta z
\]
\includegraphics[width=0.5\linewidth]{lorenz_attractor.png}
\end{frame}

\begin{frame}{Lyapunov Exponents}
\begin{itemize}
    \item \textbf{Largest Lyapunov exponent $\lambda_{\max}$:}
         \[
         \|\delta x(t)\| \approx e^{\lambda_{\max} t}\|\delta x(0)\|
         \]
    \item $\lambda_{\max} > 0$ indicates chaos.
    \item Connection to entropy; practical computation
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Selected Topics and Modern Directions}

\begin{frame}{Stochastic & High-Dimensional Systems}
\begin{itemize}
    \item \textbf{Stochastic DS:} Langevin, Fokker-Planck equations
    \item \textbf{Networks:} Synchronization, coupled oscillators (Kuramoto model)
    \item \textbf{Control, computation, and learning:}
    \begin{itemize}
      \item Neural ODEs (Machine learning, deep learning)
      \item Model reduction (e.g., Koopman operator theory)
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Numerical Methods and Simulation}
\begin{itemize}
    \item Runge-Kutta and Euler methods
    \item Event-driven integration
    \item Software: \texttt{Python/scipy.integrate}, \texttt{MATLAB ode45}, \texttt{Julia DifferentialEquations.jl}
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Further Study and References}

\begin{frame}{Further Reading}
\begin{itemize}
    \item S.H. Strogatz, \emph{Nonlinear Dynamics and Chaos} (2018)
    \item L. Perko, \emph{Differential Equations and Dynamical Systems} (2013)
    \item E. Ott, \emph{Chaos in Dynamical Systems} (2002)
    \item P. Gaspard, \emph{Chaos, Scattering, and Statistical Mechanics} (1998)
    \item J.D. Meiss, \emph{Differential Dynamical Systems} (2017)
    \item Lecture notes by Steven H. Strogatz (\url{https://courses.cit.cornell.edu/math4410/book.pdf})
\end{itemize}
\end{frame}

\begin{frame}
\centering
\Huge{\textbf{Thank you!}}
\newline
\vspace{0.5cm}
\Large{Questions? Discussion.}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
```

---

**A few notes:**

- **Images** referenced (e.g., `logmap_bifurcation.png`, `saddlenode_bifurcation.png`, `lorenz_attractor.png`) should be created or downloaded and placed in the same folder as your `.tex` file. For public domain images, see Wikimedia or generate them using Python/Matplotlib.
- You can easily extend or split this into multiple lectures, going deeper into proofs, computation, or advanced topics (e.g., measure-theoretic dynamics, ergodic theory, or PDEs).
- Feel free to edit author/institute, add more technical detail, or include worked examples as homeworks or supplementary slides.

If you want **more focused slides on a particular topic** (Lyapunov functions, bifurcation theory, etc.), let me know!

Below is a comprehensive outline and example LaTeX Beamer code for a graduate lecture series on **Dynamical Systems Theory**, tailored for a multidisciplinary audience (Physics, Applied Math, CS, ECE). The slides blend rigor and accessibility, emphasizing interdisciplinary applications.

---

### **Week 1: Introduction to Dynamical Systems**
**Slide Code**: `week1-intro.tex`
```latex
\documentclass[11pt]{beamer}
\usetheme{Berkeley}
\usepackage{amsmath, amssymb, graphicx}

\title{Dynamical Systems Theory}
\subtitle{An Interdisciplinary Graduate Introduction}
\author{Prof. [Your Name]}
\institute{Physics / Applied Math / CS / ECE}
\date{Week 1: Fundamentals}

\begin{document}

% Title Slide
\frame{\titlepage}

% Outline
\begin{frame}
\frametitle{Week 1 Outline}
\tableofcontents
\end{frame}

% Section: Definition and Examples
\section{What is a Dynamical System?}
\begin{frame}
\frametitle{Definition}
\begin{definition}
A dynamical system is a tuple $(M, f, T)$ where:
\begin{itemize}
\item $M$: Phase space (manifold)
\item $T$: Time domain ($\mathbb{R}$ or $\mathbb{Z}$)
\item $f: T \times M \rightarrow M$: Evolution rule
\end{itemize}
\end{definition}
\end{frame}

\begin{frame}
\frametitle{Examples Across Disciplines}
\begin{columns}
\column{0.5\textwidth}
\textbf{Physics}: Pendulum $$\ddot{\theta} + \sin\theta = 0$$
\column{0.5\textwidth}
\textbf{ECE}: RLC Circuits 
$$ L\ddot{q} + R\dot{q} + \frac{q}{C} = 0 $$
\end{columns}
\vfill
\textbf{CS}: Logistic Map $x_{n+1} = rx_n(1 - x_n)$ (Population/Algorithms)  
\textbf{Math}: Van der Pol Oscillator for nonlinear limit cycles.
\end{frame}

% Section: Types of Dynamics
\section{Continuous vs. Discrete}
\begin{frame}
\frametitle{Types of Systems}
\begin{description}
\item[Continuous] Described by ODEs (e.g., Hamiltonian systems)
\item[Discrete] Given by maps (e.g., weather prediction models)
\end{description}
\includegraphics[width=\textwidth]{figures/continuity.png} % Placeholder
\end{frame}

% Motivation Slide
\section{Why Study This?}
\begin{frame}
\frametitle{Motivation and Impact}
\begin{itemize}
\item Modeling the universe: From quantum chaos to neural networks
\item Predictive simulations in climate, epidemiology, and economics
\item Foundations for machine learning (recurrent neural networks)
\end{itemize}
\end{frame}

% Future Outlook
\begin{frame}
\frametitle{Future Lectures}
\begin{enumerate}
\item ODEs \& Stability Analysis \hfill (Week 2-4)
\item Bifurcations \& Chaos \hfill (Week 5-9)
\item Applications in CS/ECE \hfill (Week 10-12)
\end{enumerate}
\end{frame}

\end{document}
```

---

### **Week 6: Chaos and Lyapunov Exponents**
**Slide Code**: `week6-chaos.tex`
```latex
\documentclass[11pt]{beamer}
\usetheme{CambridgeUS}
\usepackage{tikz, pgfplots}

\title{Dynamical Systems Theory}
\subtitle{Chaos and Strange Attractors}
\author{Prof. [Your Name]}
\institute{Week 6}
\date{\today}

\begin{document}

\frame{\titlepage}

% Table of Contents
\section{Chaos Overview}
\section{Lyapunov Exponents}
\section{Strange Attractors}

\begin{frame}
\frametitle{Plan for Week 6}
\tableofcontents
\end{frame}

% Sensitivity to Initial Conditions
\section{Chaos}
\begin{frame}
\frametitle{Sensitivity and Unpredictability}
\textbf{Butterfly Effect}: Small changes in initial conditions lead to diverging outcomes.  
\[
\text{Distance Between Trajectories} \sim e^{\lambda t}
\]
\begin{block}{Definition: Chaos}
A system is chaotic if it has:
\begin{enumerate}
\item Dense periodic orbits
\item Transitivity (local mixing)
\item Sensitive dependence on ICs
\end{enumerate}
\end{block}
\end{frame}

% Lyapunov Exponents
\begin{frame}
\frametitle{Quantifying Chaos: Lyapunov Exponents}
\[
\lambda(x_0) = \lim_{t \to \infty} \lim_{\delta_0 \to 0} \frac{1}{t} \ln\left(\frac{|\delta x(t)|}{|\delta_0|}\right)
\]
\begin{itemize}
\item Positive exponent $\Rightarrow$ chaos
\item Lorenz system: $\lambda_1 \approx 0.9$, $\lambda_2 = 0$, $\lambda_3 \approx -14$
\end{itemize}
\end{frame}

% Attractors
\section{Strange Attractors}
\begin{frame}
\frametitle{Examples of Strange Attractors}
\begin{columns}
\column{0.4\textwidth}
\textbf{Lorenz Attractor}  
$$
\begin{cases}
\dot{x} = \sigma(y - x) \\
\dot{y} = x(\rho - z) - y \\
\dot{z} = xy - \beta z
\end{cases}
$$
\column{0.6\textwidth}
\begin{tikzpicture}
\pgfplotsset{width=5cm, height=5cm}
\begin{axis}[hide axis]
\addplot3[blue, samples=1000, domain=0:100, variable=\t]
({sin(\t r)}, {cos(\t r)}, {\t/30});
\end{axis}
\end{tikzpicture}
Fractal Dimension $\approx 2.06$
\end{columns}
\end{frame}

% Applications
\begin{frame}
\frametitle{Chaos in Applications}
\textbf{Computer Science}: PRNGs, cryptography.  
\textbf{Physics}: Turbulence, quantum scars.  
\textbf{ECE}: Chaotic circuits for secure communication.  
\textbf{Math}: Open problems in ergodic theory.
\end{frame}

\end{document}
```

---

### **Structural Features**
#### Slide Customization
- **Themes**: Use `Berkeley`, `CambridgeUS`, or `Madrid` for clean layouts.
- **Graphics**: Integrate `TikZ` plots/flows or `pgfplots` for visualizing attractors/(un)stable manifolds.
- **Math Environments**: Define `examples`, `definitions`, `theorems` via Beamer blocks (`block`, `exampleblock`, `alertblock`).

#### Content Highlights
- **Interdisciplinary Examples**: E.g., in bifurcations, link Hopf bifurcations to laser dynamics (Physics) and neural oscillators (CS).
- **Visual Aids**: Embed animated figures (e.g., bifurcation diagrams, Henon maps) using `animate` package.
- **Optional Sections**: Use `\onslide<2->` to progressively reveal derivations or numerical simulations.

#### Supplementary Resources
- **Project Guidance**: Include a final slide with project topics (e.g., "Apply SINDy to GPU power traces" for ECE, "Chaos in Hamiltonian Monte Carlo" for CS).
- **Bibliography**: Footnote citations (Strogatz, Katok-Hasselblatt, Ott) on relevant slides.

---

This framework ensures flexibility for expansion to other weeks, consistent notation, and adaptability across disciplines. Each lecture builds from foundational concepts to applications, fostering engagement through cross-disciplinary relevance.

Below is a LaTeX Beamer template for a graduate-level course on **Dynamical Systems Theory**, tailored for PhD students in physics, applied math, computer science, and ECE. The slides balance mathematical rigor with intuitive explanations and applications.

---

### **`dynamical-systems.tex`**
```latex
\documentclass[10pt, xcolor={table,dvipsnames}, aspectratio=169]{beamer}

% === Packages ===
\usepackage{amsmath, amssymb, amsfonts, amsthm}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{shapes, arrows, shadows, positioning, calc, fit}
\usepackage{booktabs}
\usepackage{multirow}
\usepackage{tabularx}
\usepackage{hyperref}

% === Beamer Settings ===
\usetheme{Madrid}
\useinnertheme{circles}
\setbeamercolor{title}{fg=MidnightBlue}
\setbeamercolor{frametitle}{fg=MidnightBlue}
\setbeamerfont{frametitle}{series=\bfseries}

% === Custom Commands ===
\newcommand{\diff}[2]{\frac{d #1}{d #2}}
\newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\state}[1]{\mathbf{#1}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\prob}[1]{\mathbb{P}(#1)}

% === Title Page ===
\title{Dynamical Systems Theory}
\subtitle{Lectures for Graduate Students}
\author{Dr. Instructor Name}
\institute{University Name}
\date{Spring 2024}

% === Document ===
\begin{document}

% === Title Slide ===
\begin{frame}
    \titlepage
\end{frame}

% === Outline ===
\begin{frame}{Outline}
    \begin{enumerate}
        \item \textbf{Introduction}: Definitions, Classifications, and Examples
        \item \textbf{Linear Dynamical Systems}: Stability, Matrix Exponentials
        \item \textbf{Nonlinear Dynamics}: Fixed Points, Linearization, Bifurcations
        \item \textbf{Lyapunov Stability Theory}
        \item \textbf{Chaos and Strange Attractors}
        \item \textbf{Applications} (Physics, CS, ECE)
    \end{enumerate}
\end{frame}

% === Introduction ===
\section{Introduction}
\begin{frame}{What is a Dynamical System?}
    \textbf{Definition}: A dynamical system is a tuple $(T, X, \Phi)$ where:
    \begin{itemize}
        \item $T$: Time domain (\textbf{discrete} $\mathbb{Z}$ or \textbf{continuous} $\mathbb{R}$)
        \item $X$: State space ($\mathbb{R}^n$, manifolds, functional spaces)
        \item $\Phi: T \times X \to X$: Evolution function (e.g., ODEs, maps)
    \end{itemize}
    
    \begin{exampleblock}{Examples}
        \begin{itemize}
            \item Pendulum: $\ddot{\theta} + \gamma \dot{\theta} + \frac{g}{L} \sin \theta = 0$
            \item Lorenz system: $\dot{x} = \sigma(y - x), \dot{y} = x(\rho - z) - y, \dot{z} = xy - \beta z$
            \item Logistic map: $x_{n+1} = r x_n (1 - x_n)$ (discrete chaos)
        \end{itemize}
    \end{exampleblock}
\end{frame}

% === Linear Systems ===
\section{Linear Dynamical Systems}
\begin{frame}{Linear ODEs}
    General form: $\dot{\state{x}} = A\state{x}, \quad \state{x} \in \R^n, A \in \R^{n \times n}$
    
    \textbf{Solution}:
    \[
    \state{x}(t) = e^{At} \state{x}_0
    \]
    where $e^{At}$ is the \textbf{matrix exponential}.
    
    \begin{block}{Stability Criteria}
        \begin{itemize}
            \item Eigenvalues $\lambda_j$ of $A$:
            \[
            \text{Re}(\lambda_j) < 0 \ \forall j \implies \text{Asymptotically stable}
            \]
            \item Jordan blocks determine transient behavior.
        \end{itemize}
    \end{block}
\end{frame}

% === Nonlinear Dynamics ===
\section{Nonlinear Dynamics}
\begin{frame}{Fixed Points and Linearization}
    For $\dot{\state{x}} = f(\state{x})$:
    \begin{enumerate}
        \item Find fixed points $\state{x}^*$ where $f(\state{x}^*) = 0$.
        \item Linearize:
        \[
        \dot{\state{y}} = J(\state{x}^*) \state{y}, \quad J = \left. \pdiff{f_i}{x_j} \right|_{\state{x}^*}
        \]
    \end{enumerate}
    
    \textbf{Bifurcations}:
    \begin{figure}
        \centering
        \includegraphics[width=0.5\textwidth]{bifurcation_diagram.png}
        \caption{Saddle-node, Hopf, pitchfork bifurcations.}
    \end{figure}
\end{frame}

% === Lyapunov Theory ===
\section{Lyapunov Stability}
\begin{frame}{Lyapunov Functions}
    \textbf{Theorem} (Lyapunov, 1892): If $\exists V(\state{x})$ such that:
    \[
    \begin{cases}
        V(\state{x}) > 0 \quad \forall \state{x} \neq 0, \\
        \dot{V}(\state{x}) = \nabla V \cdot f(\state{x}) \leq 0,
    \end{cases}
    \]
    then $\state{x} = 0$ is stable. If $\dot{V} < 0$, asymptotically stable.
    
    \begin{exampleblock}{Example: Pendulum with Friction}
        \small
        $\theta'' + \gamma \theta' + \sin \theta = 0 \implies$ Use $V(\theta, \omega) = 1 - \cos \theta + \frac{\omega^2}{2}$.
    \end{exampleblock}
\end{frame}

% === Chaos ===
\section{Chaos}
\begin{frame}{Defining Chaos}
    \textbf{Properties}:
    \begin{itemize}
        \item Sensitive dependence on initial conditions (Lyapunov exponents $> 0$).
        \item Topological mixing.
        \item Dense periodic orbits.
    \end{itemize}
    
    \textbf{Examples}:
    \begin{itemize}
        \item Lorenz attractor (weather models).
        \item Double pendulum (Hamiltonian chaos).
        \item Logistic map ($r = 4$: fully chaotic).
    \end{itemize}
    
    \begin{tikzpicture}[overlay]
        \node at (11, 2) {\includegraphics[width=3cm]{lorenz_attractor.png}};
    \end{tikzpicture}
\end{frame}

% === Applications ===
\section{Applications}
\begin{frame}{Interdisciplinary Examples}
    \begin{table}
        \centering
        \begin{tabular}{ll}
            \toprule
            \textbf{Field} & \textbf{Application} \\
            \midrule
            Physics & Celestial mechanics, Quantum chaos \\
            CS & Recurrent neural networks, Algorithm stability \\
            ECE & Circuit dynamics, Control theory \\
            Biology & Population dynamics, Neural oscillators \\
            \bottomrule
        \end{tabular}
    \end{table}
    
    \begin{block}{Open Research}
        \begin{itemize}
            \item Machine learning for high-dimensional dynamics.
            \item Control of chaotic systems (e.g., stabilizing turbulence).
        \end{itemize}
    \end{block}
\end{frame}

% === Final Slide ===
\begin{frame}{Summary and References}
    \textbf{Key Takeaways}:
    \begin{itemize}
        \item Dynamical systems unify continuous/discrete modeling.
        \item Stability and bifurcations are central concepts.
        \item Chaos appears in deterministic systems.
    \end{itemize}
    
    \textbf{Further Reading}:
    \begin{itemize}
        \item Strogatz, \textit{Nonlinear Dynamics and Chaos}
        \item Wiggins, \textit{Introduction to Applied Nonlinear Dynamical Systems}
    \end{itemize}
\end{frame}

\end{document}
```

---

### **Key Features of the Slide Deck**
1. **Mathematical Rigor**: Uses clear definitions, theorems, and derivations.
2. **Interdisciplinary Examples**: Covers physics (pendulum, Lorenz), CS (chaotic maps), ECE (circuits).
3. **Visual Aids**: Diagrams for bifurcations, attractors, and stability regions.
4. **Structured Flow**: From linear → nonlinear → chaos → applications.
5. **Beamer Enhancements**: Color themes, overlays, and custom commands for readability.

### **How to Use**
- Replace placeholder images (`bifurcation_diagram.png`, `lorenz_attractor.png`) with your own.
- Add more slides as needed (e.g., exercises, proofs, case studies).
- Customize the theme (`Madrid`) and colors (`MidnightBlue`).

Let me know if you'd like additional slides on specific topics!

Below is a self‐contained LaTeX Beamer source for a multi‐lecture graduate course on Dynamical Systems Theory.  You can compile it with any modern LaTeX distribution (e.g. TeX Live, MikTeX).  Just save as `dynamics_course.tex` and run `pdflatex` (twice).

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

\documentclass[11pt]{beamer}
\usetheme{Madrid}
\usecolortheme{dolphin}
\usefonttheme{professionalfonts}

\usepackage{amsmath,amssymb,graphicx}
\usepackage{tikz}
\setbeamertemplate{caption}[numbered]

\title[Dynamical Systems Theory]{Graduate Course on Dynamical Systems Theory}
\author{Instructor: Your Name}
\institute[Your Univ.]{Department of Physics \& Applied Math \\ Your University}
\date{\today}

\begin{document}

% Title page
\begin{frame}
  \titlepage
\end{frame}

% Outline
\begin{frame}{Outline}
  \tableofcontents
\end{frame}

% ------------------------------------------------------------------------
\section{Introduction}
% ------------------------------------------------------------------------

\begin{frame}{What Is a Dynamical System?}
  \begin{itemize}
    \item A rule describing time‐evolution of a state $x(t)$ in a state‐space $X$.
    \item Continuous‐time: $\dot x = f(x,t)$,  $x\in\mathbb{R}^n$.
    \item Discrete‐time (maps): $x_{k+1}=F(x_k,k)$.
    \item Autonomous vs.\ non‐autonomous; deterministic vs.\ stochastic.
  \end{itemize}
\end{frame}

\begin{frame}{Examples Across Disciplines}
  \begin{itemize}
    \item Classical mechanics: Hamilton’s equations.
    \item Electrical circuits: RLC networks, power‐system dynamics.
    \item Population biology: Lotka–Volterra, epidemiology.
    \item Control theory: state‐space models.
    \item Computer science: neural networks, agent‐based models.
  \end{itemize}
  \vspace{1ex}
  \begin{center}
    \includegraphics[height=2cm]{example_phase_plane.pdf}\\
    {\footnotesize Phase portrait of a 2D system}
  \end{center}
\end{frame}

% ------------------------------------------------------------------------
\section{Linear Systems}
% ------------------------------------------------------------------------

\begin{frame}{Continuous‐Time Linear Systems}
  \[
    \dot x = A\,x,\quad A\in\mathbb{R}^{n\times n}.
  \]
  Solution: $x(t)=e^{At}x(0)$.
  \begin{itemize}
    \item Eigenvalues $\{\lambda_i\}$ determine stability:
      Re$(\lambda_i)<0\Rightarrow$ asymptotic stability.
    \item Modal decomposition if $A$ diagonalizable.
  \end{itemize}
\end{frame}

\begin{frame}{Discrete‐Time Linear Maps}
  \[
    x_{k+1} = A\,x_k.
  \]
  Solution: $x_k = A^k x_0$.
  \begin{itemize}
    \item Spectrum $\{\lambda_i\}$: $|\lambda_i|<1$ $\Rightarrow$ stable fixed point.
    \item Jordan forms, companion matrices.
  \end{itemize}
\end{frame}

\begin{frame}{Example: 2×2 System}
  \[
    A = \begin{pmatrix}a & b \\ c & d\end{pmatrix},\quad
    \lambda_{1,2} = \frac{a+d\pm\sqrt{(a-d)^2+4bc}}{2}.
  \]
  \begin{itemize}
    \item Node, focus, saddle, center.
    \item Trace–Determinant plane classification.
  \end{itemize}
  \begin{center}
    \includegraphics[height=2cm]{trace_det_plane.pdf}
  \end{center}
\end{frame}

% ------------------------------------------------------------------------
\section{Phase‐Plane Analysis}
% ------------------------------------------------------------------------

\begin{frame}{Autonomous 2D Systems}
  \[
    \dot x = f(x,y),\quad \dot y = g(x,y).
  \]
  \begin{itemize}
    \item Nullclines: $f=0$, $g=0$. Intersection = equilibria.
    \item Linearize: $J=\begin{pmatrix}f_x & f_y\\g_x & g_y\end{pmatrix}$.
  \end{itemize}
\end{frame}

\begin{frame}{Classification of Fixed Points}
  \begin{itemize}
    \item Real distinct eigenvalues: node (stable/unstable).
    \item Complex conjugate: focus/spiral.
    \item Opposite signs: saddle.
    \item Pure imaginary: center (linearization inconclusive).
  \end{itemize}
  \begin{center}
    \includegraphics[height=3cm]{phase_portraits.pdf}
  \end{center}
\end{frame}

\begin{frame}{Example: Nonlinear Oscillator}
  Van der Pol:
  \[
    \ddot x - \mu (1 - x^2)\dot x + x = 0,
    \quad
    \begin{cases}
      \dot x = y,\\
      \dot y = \mu(1-x^2)y - x.
    \end{cases}
  \]
  \begin{itemize}
    \item Limit cycle for $\mu>0$.
    \item Energy pumping/dissipation.
  \end{itemize}
\end{frame}

% ------------------------------------------------------------------------
\section{Nonlinear Systems \& Stability}
% ------------------------------------------------------------------------

\begin{frame}{Linearization and Hartman–Grobman}
  \begin{itemize}
    \item If $x^*$ s.t.\ $f(x^*)=0$ and $J=Df(x^*)$ has no eigenvalues on imaginary axis, 
    then local behavior of $\dot x=f(x)$ is topologically conjugate to $\dot y=Jy$.
  \end{itemize}
\end{frame}

\begin{frame}{Lyapunov Stability Definitions}
  Equilibrium $x^*$:
  \begin{itemize}
    \item Stable: $\forall\varepsilon>0\,\exists\delta\!: \|x(0)-x^*\|<\delta\Rightarrow \|x(t)-x^*\|<\varepsilon\,\forall t$.
    \item Asymptotically stable: stable + $\lim_{t\to\infty}x(t)=x^*$.
    \item Unstable: not stable.
  \end{itemize}
\end{frame}

% ------------------------------------------------------------------------
\section{Lyapunov Methods}
% ------------------------------------------------------------------------

\begin{frame}{Lyapunov’s Direct Method}
  Seek $V(x)\colon\mathbb{R}^n\to\mathbb{R}_{\ge0}$ with
  \[
    V(0)=0,\; V(x)>0\;(x\ne0),\quad \dot V = \nabla V\cdot f(x)\le0.
  \]
  Then $x^*=0$ is stable; if $\dot V<0$, asymptotically stable.
\end{frame}

\begin{frame}{LaSalle’s Invariance Principle}
  If $\dot V\le0$ and the largest invariant set in $\{x:\dot V=0\}$ is $\{0\}$,
  then $x(t)\to0$ as $t\to\infty$.
  \vspace{2ex}
  \begin{itemize}
    \item Powerful for mechanical systems (energy‐based $V$).
    \item Extensions to non‐autonomous and discrete‐time.
  \end{itemize}
\end{frame}

% ------------------------------------------------------------------------
\section{Bifurcation Theory}
% ------------------------------------------------------------------------

\begin{frame}{One‐Parameter Bifurcations}
  Consider $\dot x = f(x,\mu)$.
  \begin{itemize}
    \item Saddle‐node: two equilibria collide and annihilate.
    \item Transcritical, pitchfork: exchange/stability change.
    \item Hopf: fixed point $\to$ small‐amplitude limit cycle.
  \end{itemize}
\end{frame}

\begin{frame}{Example: Saddle‐Node}
  \[
    \dot x = \mu - x^2,\;
    \begin{cases}
      \mu<0: \text{no equilibria},\\
      \mu=0: \text{one degenerate},\\
      \mu>0: x=\pm\sqrt\mu.
    \end{cases}
  \]
  \includegraphics[height=2cm]{saddle_node_bif.pdf}
\end{frame}

\begin{frame}{Hopf Bifurcation}
  \[
    \dot z = (\mu + i\omega)z - |z|^2 z,\quad z\in\mathbb{C}.
  \]
  \begin{itemize}
    \item At $\mu=0$: Re$(\lambda)=0$, cycle born for $\mu>0$ (supercritical).
    \item Amplitude $\sqrt\mu$, frequency $\omega$.
  \end{itemize}
\end{frame}

% ------------------------------------------------------------------------
\section{Chaos \& Strange Attractors}
% ------------------------------------------------------------------------

\begin{frame}{Discrete Chaos: Logistic Map}
  \[
    x_{k+1} = r\,x_k(1 - x_k),\quad 0\le x_k\le1.
  \]
  \begin{itemize}
    \item Period‐doubling route to chaos as $r$ increases.
    \item Lyapunov exponent $\lambda = \lim \frac1N\sum\ln|f'(x_k)|$.
  \end{itemize}
  \includegraphics[height=2cm]{logistic_bifurcation.pdf}
\end{frame}

\begin{frame}{Continuous Chaos: Lorenz System}
  \[
    \begin{cases}
      \dot x = \sigma(y - x),\\
      \dot y = x(\rho - z) - y,\\
      \dot z = xy - \beta z.
    \end{cases}
  \]
  Sensitive dependence, strange attractor for canonical parameters.
  \includegraphics[height=2.5cm]{lorenz_attractor.pdf}
\end{frame}

\begin{frame}{Characterizing Chaos}
  \begin{itemize}
    \item Lyapunov spectrum.
    \item Poincaré sections.
    \item Fractal (Hausdorff) dimension.
    \item Entropy measures.
  \end{itemize}
\end{frame}

% ------------------------------------------------------------------------
\section{Numerical Methods}
% ------------------------------------------------------------------------

\begin{frame}{Time–Integration Schemes}
  \begin{itemize}
    \item Euler, Heun, Runge–Kutta 4.
    \item Stiff solvers: implicit methods, BDF.
    \item Step‐size control, error estimation.
  \end{itemize}
\end{frame}

\begin{frame}{Continuation \& Bifurcation Software}
  \begin{itemize}
    \item AUTO, MATCONT, PyDSTool.
    \item Compute equilibrium branches, cycles, detection of bifurcations.
    \item High‐dimensional continuation (e.g.\ PDE discretizations).
  \end{itemize}
\end{frame}

% ------------------------------------------------------------------------
\section{Applications}
% ------------------------------------------------------------------------

\begin{frame}{Control Systems}
  \begin{itemize}
    \item Feedback linearization, sliding mode, Lyapunov‐based design.
    \item Robustness, H–infinity, passivity.
  \end{itemize}
\end{frame}

\begin{frame}{Neuroscience \& Networks}
  \begin{itemize}
    \item Hodgkin–Huxley models, neural mass models.
    \item Synchronization, consensus, epidemic spreading.
    \item Complex networks: stability, percolation.
  \end{itemize}
\end{frame}

\begin{frame}{Mechanical and Power Systems}
  \begin{itemize}
    \item Nonlinear oscillators, energy harvesting.
    \item Swing equation in power grids, small‐signal stability.
    \item Transient stability, bifurcation to voltage collapse.
  \end{itemize}
\end{frame}

% ------------------------------------------------------------------------
\section{Advanced Topics \& Outlook}
% ------------------------------------------------------------------------

\begin{frame}{Infinite‐Dimensional Systems}
  \begin{itemize}
    \item Delay differential equations.
    \item Partial differential equations (reaction–diffusion, Navier–Stokes).
    \item Center manifold, inertial manifolds.
  \end{itemize}
\end{frame}

\begin{frame}{Stochastic Dynamical Systems}
  \begin{itemize}
    \item SDEs: Ito vs.\ Stratonovich.
    \item Random dynamical systems, ergodic theory.
    \item Noise‐induced transitions, large deviations.
  \end{itemize}
\end{frame}

% ------------------------------------------------------------------------
\begin{frame}{References \& Further Reading}
  \begin{itemize}
    \item S. Strogatz, {\em Nonlinear Dynamics and Chaos}.
    \item H. Khalil, {\em Nonlinear Systems}.
    \item P. G. Drazin, {\em Nonlinear Systems}.
    \item J. Guckenheimer \& P. Holmes, {\em Nonlinear Oscillations}.
    \item L. Perko, {\em Differential Equations and Dynamical Systems}.
  \end{itemize}
\end{frame}

\begin{frame}{Thank You / Questions?}
  \centering\vspace{4ex}
  {\Large Thank you!}\\
  {\small Contact: your.email@university.edu}
\end{frame}

\end{document}

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

Instructions:
1. Replace “Your Name”, “Your Univ.”, and contact info.
2. Add or swap figures (place PDFs/PNGs in the same directory).
3. Extend any section or split into separate Beamer files per lecture if desired.
4. Compile with `pdflatex dynamics_course.tex`.

Here is a comprehensive set of lecture slides on dynamical systems theory using LaTeX Beamer, tailored for PhD students from physics, applied math, computer science, and ECE.

**Lecture 1: Introduction to Dynamical Systems**

```latex
\documentclass{beamer}
\usetheme{default}

\begin{document}

\begin{frame}
  \frametitle{Introduction to Dynamical Systems}
  \begin{itemize}
    \item Dynamical systems: study of systems that change over time
    \item Examples: population dynamics, chemical reactions, electrical circuits, climate models
    \item Key concepts: state space, trajectories, attractors, bifurcations
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Types of Dynamical Systems}
  \begin{itemize}
    \item Continuous-time vs. discrete-time
    \item Linear vs. nonlinear
    \item Autonomous vs. non-autonomous
    \item Deterministic vs. stochastic
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Mathematical Representations}
  \begin{itemize}
    \item Ordinary differential equations (ODEs)
    \item Difference equations
    \item State-space models
  \end{itemize}
\end{frame}

\end{document}
```

**Lecture 2: Linear Dynamical Systems**

```latex
\documentclass{beamer}
\usetheme{default}

\begin{document}

\begin{frame}
  \frametitle{Linear Dynamical Systems}
  \begin{itemize}
    \item Definition: $\dot{x} = Ax$, where $A$ is a matrix
    \item Solution: $x(t) = e^{At}x(0)$
    \item Properties: stability, controllability, observability
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Stability Analysis}
  \begin{itemize}
    \item Eigenvalues of $A$ determine stability
    \item Lyapunov stability theory
    \item Examples: harmonic oscillator, damped oscillator
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Controllability and Observability}
  \begin{itemize}
    \item Controllability: can we steer the system to a desired state?
    \item Observability: can we infer the state from output measurements?
    \item Kalman rank condition, Gramian
  \end{itemize}
\end{frame}

\end{document}
```

**Lecture 3: Nonlinear Dynamical Systems**

```latex
\documentclass{beamer}
\usetheme{default}

\begin{document}

\begin{frame}
  \frametitle{Nonlinear Dynamical Systems}
  \begin{itemize}
    \item Definition: $\dot{x} = f(x)$, where $f$ is nonlinear
    \item Examples: logistic map, Lorenz system, pendulum
    \item Properties: bifurcations, chaos, attractors
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Bifurcations}
  \begin{itemize}
    \item Qualitative changes in behavior as parameters vary
    \item Types: pitchfork, Hopf, saddle-node
    \item Examples: logistic map, Van der Pol oscillator
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Chaos}
  \begin{itemize}
    \item Sensitive dependence on initial conditions
    \item Lyapunov exponents, entropy
    \item Examples: Lorenz system, logistic map
  \end{itemize}
\end{frame}

\end{document}
```

**Lecture 4: Attractors and Invariant Manifolds**

```latex
\documentclass{beamer}
\usetheme{default}

\begin{document}

\begin{frame}
  \frametitle{Attractors}
  \begin{itemize}
    \item Definition: a set that attracts nearby trajectories
    \item Types: fixed points, limit cycles, strange attractors
    \item Examples: Lorenz system, Van der Pol oscillator
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Invariant Manifolds}
  \begin{itemize}
    \item Definition: a manifold that is invariant under the flow
    \item Types: stable, unstable, center manifolds
    \item Examples: linear systems, nonlinear systems
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Stable and Unstable Manifolds}
  \begin{itemize}
    \item Definition: manifolds that are tangent to the stable/unstable eigenspaces
    \item Properties: invariance, smoothness
    \item Examples: linear systems, nonlinear systems
  \end{itemize}
\end{frame}

\end{document}
```

**Lecture 5: Bifurcation Theory**

```latex
\documentclass{beamer}
\usetheme{default}

\begin{document}

\begin{frame}
  \frametitle{Bifurcation Theory}
  \begin{itemize}
    \item Study of qualitative changes in behavior as parameters vary
    \item Types: local, global bifurcations
    \item Examples: pitchfork, Hopf, saddle-node bifurcations
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Local Bifurcations}
  \begin{itemize}
    \item Changes in behavior near a fixed point
    \item Types: pitchfork, Hopf, saddle-node
    \item Examples: logistic map, Van der Pol oscillator
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Global Bifurcations}
  \begin{itemize}
    \item Changes in behavior that are not local to a fixed point
    \item Types: homoclinic, heteroclinic bifurcations
    \item Examples: Lorenz system, Duffing oscillator
  \end{itemize}
\end{frame}

\end{document}
```

**Lecture 6: Chaos Theory**

```latex
\documentclass{beamer}
\usetheme{default}

\begin{document}

\begin{frame}
  \frametitle{Chaos Theory}
  \begin{itemize}
    \item Study of complex, aperiodic behavior in deterministic systems
    \item Characteristics: sensitive dependence on initial conditions, Lyapunov exponents
    \item Examples: Lorenz system, logistic map
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Lyapunov Exponents}
  \begin{itemize}
    \item Measure of the rate of divergence of nearby trajectories
    \item Positive Lyapunov exponent indicates chaos
    \item Examples: Lorenz system, logistic map
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Entropy and Chaos}
  \begin{itemize}
    \item Measure of the complexity of a dynamical system
    \item Kolmogorov-Sinai entropy
    \item Examples: Lorenz system, logistic map
  \end{itemize}
\end{frame}

\end{document}
```

These lecture slides cover the basics of dynamical systems theory, including linear and nonlinear systems, bifurcations, chaos, and attractors. They are designed to be accessible to PhD students from physics, applied math, computer science, and ECE. You can modify and expand them as needed to fit your specific course goals and content.

Below is a LaTeX Beamer template for the opening lectures of a graduate-level Dynamical Systems Theory course. The slides are tailored for PhD students in Physics, Applied Math, CS, and ECE, balancing mathematical rigor, physical intuition, and computational perspectives.

```latex
\documentclass[10pt,t]{beamer}
\usetheme{Madrid}
\usecolortheme{dolphin}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{physics}
\usepackage{tabularx}
\usepackage{graphicx}
\usepackage{booktabs}
\usepackage{tikz}
\usetikzlibrary{arrows.meta, shapes}
\usepackage{pifont} % For checkmarks and crosses

% Metadata
\title[Dynamical Systems Theory]{Introduction to Dynamical Systems Theory}
\subtitle{PhD Course - Physics, Applied Math, CS, ECE}
\author[Inst.]{Instructor Name}
\institute{University Name}
\date{\today}

% Custom Commands
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\fixpoint}{{\scriptstyle \square}} % Fixed-point indicator
\newcommand{\state}{{\mathbf{x}}}
\newcommand{\bmat}{\begin{bmatrix}}
\newcommand{\emat}{\end{bmatrix}}
\DeclareMathOperator{\sgn}{sgn}

\begin{document}

% Title Slide
\begin{frame}
    \titlepage
    \centering
    \includegraphics[height=0.8cm]{university_logo.png}\\ % Add university logo
    \vspace{0.2cm}
    \footnotesize We'll cover: Dynamics $\to$ Stability $\to$ Bifurcations $\to$ Chaos $\to$ Applications
\end{frame}

% Table of Contents
\begin{frame}{Roadmap}
    \tableofcontents
\end{frame}

% Core Content
\section{What are Dynamical Systems?}
\begin{frame}{Defining Dynamical Systems}
    A \alert{dynamical system} describes how a state evolves deterministically over time.  
    \vspace{1em}
    
    \begin{block}{Two Formalisms}
        \begin{itemize}
            \item \alert{Continuous-Time:} Governed by ODEs \(\Rightarrow \dv{\state}{t} = \mathbf{F}(\state, t; \boldsymbol{\mu}) \)
            \begin{itemize}
                \item \emph{Physics/ECE}: Dynamics of particles/circuits
                \item \emph{AppMath}: Phase-space analysis
            \end{itemize}
            \item \alert{Discrete-Time:} Governed by maps \(\Rightarrow \state_{t+1} = \mathbf{G}(\state_t; \boldsymbol{\mu})\)
            \begin{itemize}
                \item \emph{CS:} Iterative algorithms, network dynamics
                \item \emph{All}: Poincaré sections, time-sampling
            \end{itemize}
        \end{itemize}
    \end{block}

    \begin{center}
        \begin{tikzpicture}[scale=0.8]
            \draw[->, thick] (0,0) -- (5,0) node[right] {Time};
            \draw[->, thick] (0,0) -- (0,3) node[above] {State};
            \draw[blue, thick, smooth] plot coordinates {(0,0.5) (1,1.8) (2,1.2) (3,2.5) (4,1.0)};
            \node at (5.5,1) {$\approx$ Solutions of $d\state/dt = \mathbf{F}$};
        \end{tikzpicture}
    \end{center}
\end{frame}

\begin{frame}{Key Mathematical Ingredients}
    \begin{columns}
        \column{0.5\textwidth}
        \begin{exampleblock}{Continuous-Time}
            \[ \dot{\state} = \mathbf{F}(\state), \; \state \in \R^n \]
            \begin{itemize}
                \item Vector field $\mathbf{F}$ defines flow $\phi_t: \state(0) \mapsto \state(t)$
                \item \textcolor{blue}{Examples:}
                \begin{itemize}
                    \item Harmonic Oscillator: $\ddot{x} + \mu \dot{x} + \omega^2 x = 0$
                    \item Lorenz System: $\begin{cases} \dot{x} = \sigma(y-x) \\ \dot{y} = x(\rho - z) - y \\ \dot{z} = xy - \beta z \end{cases}$
                \end{itemize}
            \end{itemize}
        \end{exampleblock}

        \column{0.5\textwidth}
        \begin{exampleblock}{Discrete-Time}
            \[ \state_{n+1} = \mathbf{G}(\state_n), \; \state \in \mathbb{R}^n \]
            \begin{itemize}
                \item Map $\mathbf{G}$ generates sequences $\{\state_0, \state_1, \ldots \}$
                \item \textcolor{blue}{Examples:}
                \begin{itemize}
                    \item Logistic Map: $x_{n+1} = r x_n (1 - x_n)$
                    \item Hénon Map: $\begin{cases} x_{n+1} = 1 - a x_n^2 + y_n \\ y_{n+1} = b x_n \end{cases}$
                \end{itemize}
            \end{itemize}
        \end{exampleblock}
    \end{columns}
    \vspace{1em}
    \centering
    \textbf{Parameters $\boldsymbol{\mu}$ critically alter behavior!}
\end{frame}

\section{Phase Space \& First Examples}
\begin{frame}{Phase Space: The Universe of States}
    \begin{itemize}
        \item Geometric arena where dynamics "live" (e.g., $(\textcolor{blue}{x},\textcolor{red}{\dot{x}})$ for pendulum)
        \item Trajectories = Curves/sets in phase space
    \end{itemize}
    
    \begin{columns}
        \column{0.5\textwidth}
        \centering
        \textbf{Damped Pendulum}\\
        $\ddot{\theta} + \gamma \dot{\theta} + \sin\theta = 0$\\
        \vspace{0.5em}
        \includegraphics[width=0.9\textwidth]{pendulum_phase.png} % Add image of pendulum phase portrait
        
        \column{0.5\textwidth}
        \centering
        \textbf{Lorenz Attractor (Chaotic)}\\
        \vspace{0.5em}
        \includegraphics[width=0.9\textwidth]{lorenz_attractor.png} % Add image of Lorenz attractor
    \end{columns}
\end{frame}

\begin{frame}{Iterated Maps: Algorithms \& Chaos}
    \begin{block}{Logistic Map (Population Dynamics)}
        \[ x_{n+1} = r x_n (1 - x_n), \quad x_n \in [0,1] \]
        \begin{itemize}
            \item \textcolor{blue}{$r < 1$}: Extinction
            \item \textcolor{blue}{$r \in [3,4]$}: Period doubling $\to$ chaos
            \item \textcolor{blue}{CS relevance}: Pseudorandom generators
        \end{itemize}
    \end{block}
    
    \centering
    \begin{tikzpicture}[scale=0.6]
    \draw[->] (0,0) -- (10,0) node[right] {$r$};
    \draw[->] (0,0) -- (0,6) node[above] {$x^{*}$};
    \foreach \r in {0,1,2,...,10} {
        \foreach \x in {0.1,0.2,...,1} {
            \pgfmathsetmacro{\xnext}{\r*\x*(1-\x)}
            \draw[gray!30, dotted] (\r/0.5, \x*5) -- (\r/0.5 + 0.2, \xnext*5);
        }
    }
    \draw[red, thick] plot file {bif.dat}; % Precomputed bifurcation data
    \end{tikzpicture}
    
    \vspace{0.5em}
    \textbf{Lesson}: Simple rules $\neq$ simple outcomes!
\end{frame}

\section{Equilibrium Analysis \& Linearization}
\begin{frame}{Fixed Points \& Linear Stability}
    \begin{itemize}
        \item Fixed points: $\state^*$ where $\mathbf{F}(\state^*) = 0$ (continuous) or $\mathbf{G}(\state^*) = \state^*$ (discrete)
        \item \textcolor{blue}{Key question}: What happens near $\state^*$?
    \end{itemize}
    
    \begin{block}{Linearization = 1st-Order Taylor Expansion}
        Continuous: $\delta\dot{\state} = J(\state^*) \delta \state$ \quad ($J = \text{Jacobian}$)\\[0.5em]
        Stability determined by $\lambda_i$ in $\text{det}(J - \lambda I) = 0$:
        \begin{itemize}
            \item Re($\lambda$) < 0 for all $i$ $\Rightarrow$ \alert{stable}
            \item $\exists$ Re($\lambda$) > 0 $\Rightarrow$ \alert{unstable}
        \end{itemize}
    \end{block}
    
    \textcolor{blue}{Physics intuition}: Linearized dynamics are "tangent space" approximations.\\
    \textcolor{blue}{CS/ECE}: Eigenvalues $\approx$ transfer function poles.
\end{frame}

\begin{frame}{Eigenvalues Tell the Story (2D Examples)}
    \centering
    \begin{tabular}{ccc}
        \includegraphics[width=0.25\textwidth]{saddle.png} & 
        \includegraphics[width=0.25\textwidth]{node.png} & 
        \includegraphics[width=0.25\textwidth]{spiral.png} \\
        $\lambda_{1,2}$ real, $\sgn(\lambda_1) \neq \sgn(\lambda_2)$ & 
        $\lambda_{1,2}$ real, $\lambda_i < 0$ & 
        $\lambda_{1,2} = a \pm ib$, $a < 0$ \\
        \textbf{Saddle} & \textbf{Stable Node} & \textbf{Stable Spiral} \\
        (Unstable) & (Asymptotic) & (Oscillatory decay) \\
    \end{tabular}
    
    \vspace{1em}
    \textcolor{blue}{Discrete systems}: Replace Re($\lambda$) with $|\lambda| < 1$.
\end{frame}

\section{Nonlinear Phenomena Preview}
\begin{frame}{Beyond Linearity: Critical Transitions}
    \begin{columns}
        \column{0.6\textwidth}
        \begin{itemize}
            \item \textcolor{blue}{Bifurcations}: Qualitative changes at critical $\mu$
            \item \textcolor{blue}{Hopf Bifurcation}: Spiral $\rightleftharpoons$ limit cycle
            \[ \begin{cases} 
                \dot{x} = \mu x - \omega y - x(x^2+y^2) \\
                \dot{y} = \omega x + \mu y - y(x^2+y^2)
            \end{cases} \]
            \item \textcolor{blue}{ECE/Physics}: Oscillator onset, laser threshold
            \item \textcolor{blue}{Algorithmics}: Stability loss in optimization
        \end{itemize}
        
        \column{0.4\textwidth}
        \includegraphics[width=\textwidth]{hopf.png} % Hopf bifurcation diagram
    \end{columns}
    
    \vspace{1em}
    \centering
    \begin{tikzpicture}[scale=0.8]
        \draw[->] (0,0) -- (4.5,0) node[right] {$\mu$};
        \draw[->] (0,0) -- (0,3) node[above] {$||\state||$};
        \draw[red, dashed, thick] (0,0) -- (2.5,0);
        \draw[red, thick] (2.5,0) .. controls (3,1) and (3.5,2.5) .. (4,2.8);
        \draw[fill] (2.5,0) circle (0.08);
        \node at (2.5,-0.5) {$\mu_c$};
    \end{tikzpicture}
\end{frame}

\section{Computational Aspects}
\begin{frame}{Numerics \& Simulation}
    \begin{itemize}
        \item ODE Solvers: RK4, Symplectic (Hamiltonian), Stiff methods (implicit)
        \item Map Iteration: Parallelization, GPU acceleration
        \item \textcolor{blue}{Numerical Pitfalls}:
        \begin{itemize}
            \item Divergence near instabilities
            \item Chaos $\to$ error explosion (Lyapunov exponents >0)
        \end{itemize}
    \end{itemize}
    
    \begin{exampleblock}{MATLAB/Python Examples}
        \texttt{\# Python: SciPy ODE solver}\\
        \texttt{from scipy.integrate import solve\_ivp}\\
        \texttt{solve\_ivp(lorenz, [0, 100], x0, args=(sigma, rho, beta))}
    \end{exampleblock}
    
    \textcolor{blue}{Homework}: Simulate double pendulum, find critical energies for chaos.
\end{frame}

% Closing Slide
\begin{frame}{Deeper Dives Ahead!}
    \centering
    \textbf{Coming Lectures}\\
    \vspace{1em}
    \begin{tabular}{ll}
        \toprule
        \textbf{Topic} & \textbf{Key Concepts} \\
        \midrule
        Lyapunov Stability Theory & Energy-like functions, attractors \\
        Bifurcation Analysis & Normal forms, universality \\
        Hamiltonian Systems & Conservation laws, KAM theory \\
        Chaos \& Fractals & Entropy, dimension, entropy \\
        Applications & Synchroization, neural nets, control \\
        \bottomrule
    \end{tabular}
    
    \vspace{2em}
    \textbf{Message}: Universality across pendula, lasers, algorithms, and networks!
\end{frame}

\end{document}
```

## Key Design Features

1. **Interdisciplinary Relevance**:
   - Physics/Math: Phase portraits, analytical stability
   - CS/ECE: Maps as algorithms, control applications
   - Universal examples (logistic map, Lorenz attractor)

2. **Pedagogical Elements**:
   - Visual-heavy with space for figures
   - Conceptual summaries (e.g., "Eigenvalues tell the story")
   - Explicit notation for states/parameters

3. **Technical Implementation**:
   - Beamer theme with clean layout
   - Custom commands for complex symbols
   - Integrated diagrams via TikZ or pre-rendered graphics

4. **Flexibility**:
   - Spaces for institution-specific content (logo, name)
   - Scalable via comments/macros
   - Modular structure for topic extension

Simply replace placeholder images with actual figures and compile with a LaTeX engine supporting Beamer. Each section builds intuition for subsequent mathematical formalism.

I'll create a comprehensive set of lecture slides for a graduate dynamical systems theory course. Here's a structured series of lectures:

## Lecture 1: Introduction to Dynamical Systems

```latex
\documentclass{beamer}
\usetheme{Madrid}
\usecolortheme{default}

\usepackage{amsmath,amssymb,amsfonts}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{pgfplots}

\title[Dynamical Systems]{Introduction to Dynamical Systems Theory}
\subtitle{Lecture 1: Foundations and Basic Concepts}
\author{Course Instructor}
\institute{University Name}
\date{\today}

\begin{document}

\frame{\titlepage}

\begin{frame}{Course Overview}
\tableofcontents
\end{frame}

\section{What is a Dynamical System?}

\begin{frame}{Definition of Dynamical Systems}
\begin{definition}
A \textbf{dynamical system} is a mathematical model describing the time evolution of a point in a geometrical space.
\end{definition}

\begin{block}{Mathematical Formulation}
\begin{itemize}
\item \textbf{State space} $X$: set of all possible states
\item \textbf{Time domain} $T$: usually $\mathbb{R}$ or $\mathbb{Z}$
\item \textbf{Evolution rule} $\phi: T \times X \to X$
\end{itemize}
\end{block}

\begin{alertblock}{Key Property}
$\phi(t+s, x) = \phi(t, \phi(s, x))$ (Group/Semigroup property)
\end{alertblock}
\end{frame}

\begin{frame}{Types of Dynamical Systems}
\begin{columns}
\column{0.5\textwidth}
\begin{block}{Continuous Time}
\begin{align}
\frac{dx}{dt} &= f(x,t) \\
x(t_0) &= x_0
\end{align}
\begin{itemize}
\item Ordinary Differential Equations (ODEs)
\item Partial Differential Equations (PDEs)
\end{itemize}
\end{block}

\column{0.5\textwidth}
\begin{block}{Discrete Time}
\begin{align}
x_{n+1} &= f(x_n) \\
x_0 &\text{ given}
\end{align}
\begin{itemize}
\item Maps/Iterations
\item Difference equations
\end{itemize}
\end{block}
\end{columns}

\vspace{0.5cm}
\begin{block}{Autonomous vs Non-autonomous}
\begin{itemize}
\item \textbf{Autonomous}: $f$ doesn't depend explicitly on time
\item \textbf{Non-autonomous}: $f = f(x,t)$ depends on time
\end{itemize}
\end{block}
\end{frame}

\section{Examples from Different Fields}

\begin{frame}{Physics Examples}
\begin{example}[Classical Mechanics]
Newton's second law: $m\ddot{x} = F(x, \dot{x}, t)$

Convert to first-order system:
\begin{align}
\dot{x} &= v \\
\dot{v} &= \frac{1}{m}F(x, v, t)
\end{align}
\end{example}

\begin{example}[Quantum Systems]
Time-dependent Schrödinger equation:
$$i\hbar \frac{\partial \psi}{\partial t} = H \psi$$
\end{example}

\begin{example}[Statistical Mechanics]
Langevin equation:
$$m\ddot{x} = -\gamma \dot{x} + F(x) + \xi(t)$$
\end{example}
\end{frame}

\begin{frame}{Engineering and Computer Science Examples}
\begin{example}[Control Systems]
Linear time-invariant system:
$$\dot{x} = Ax + Bu, \quad y = Cx$$
\end{example}

\begin{example}[Neural Networks]
Hopfield network:
$$\frac{dv_i}{dt} = -\frac{v_i}{\tau} + \sum_j w_{ij}\sigma(v_j) + I_i$$
\end{example}

\begin{example}[Digital Signal Processing]
Discrete filter:
$$y[n] = \sum_{k=0}^{N-1} b_k x[n-k] + \sum_{k=1}^{M} a_k y[n-k]$$
\end{example}
\end{frame}

\begin{frame}{Applied Mathematics Examples}
\begin{example}[Population Dynamics]
Lotka-Volterra equations:
\begin{align}
\dot{x} &= ax - bxy \\
\dot{y} &= -cy + dxy
\end{align}
\end{example}

\begin{example}[Epidemiology]
SIR model:
\begin{align}
\dot{S} &= -\beta SI \\
\dot{I} &= \beta SI - \gamma I \\
\dot{R} &= \gamma I
\end{align}
\end{example}
\end{frame}

\section{Phase Space and Trajectories}

\begin{frame}{Phase Space Concepts}
\begin{definition}[Phase Space]
The space of all possible states of a system, where each point represents a complete specification of the system's state.
\end{definition}

\begin{block}{Key Concepts}
\begin{itemize}
\item \textbf{Trajectory/Orbit}: $\{x(t) : t \in T\}$ for initial condition $x_0$
\item \textbf{Flow}: The collection of all trajectories
\item \textbf{Vector Field}: $f(x)$ giving direction and magnitude of motion
\end{itemize}
\end{block}

\begin{alertblock}{Uniqueness}
Under suitable conditions (Lipschitz continuity), trajectories don't intersect in autonomous systems.
\end{alertblock}
\end{frame}

\begin{frame}{Geometric Perspective}
\begin{center}
\begin{tikzpicture}[scale=0.8]
% Phase portrait example
\begin{axis}[
    axis lines = center,
    xlabel = $x_1$,
    ylabel = $x_2$,
    domain = -3:3,
    samples = 20,
    xmin=-3, xmax=3,
    ymin=-3, ymax=3,
]

% Vector field arrows
\addplot3[blue, quiver={u={-y}, v={x}, scale arrows=0.3}] 
    {0};

% Sample trajectories (circles for this example)
\addplot[red, thick, domain=0:360] ({cos(x)}, {sin(x)});
\addplot[red, thick, domain=0:360] ({1.5*cos(x)}, {1.5*sin(x)});
\addplot[red, thick, domain=0:360] ({2.5*cos(x)}, {2.5*sin(x)});

\end{axis}
\end{tikzpicture}
\end{center}
Simple harmonic oscillator: $\dot{x} = y, \dot{y} = -x$
\end{frame}

\section{Equilibria and Fixed Points}

\begin{frame}{Equilibrium Points}
\begin{definition}[Equilibrium/Fixed Point]
For continuous systems: $x^*$ such that $f(x^*) = 0$

For discrete systems: $x^*$ such that $f(x^*) = x^*$
\end{definition}

\begin{block}{Types of Equilibria}
\begin{itemize}
\item \textbf{Stable}: Small perturbations decay
\item \textbf{Unstable}: Small perturbations grow
\item \textbf{Asymptotically stable}: Stable + attracting
\item \textbf{Neutrally stable}: Bounded but don't decay
\end{itemize}
\end{block}

\begin{example}
$\dot{x} = -x^3$
\begin{itemize}
\item Equilibrium at $x^* = 0$
\item $f'(0) = 0$ (linearization inconclusive)
\item Globally asymptotically stable
\end{itemize}
\end{example}
\end{frame}

\section{Linear Systems Analysis}

\begin{frame}{Linear Systems}
\begin{block}{Continuous Linear System}
$$\dot{x} = Ax, \quad x(0) = x_0$$

Solution: $x(t) = e^{At}x_0$
\end{block}

\begin{block}{Discrete Linear System}
$$x_{n+1} = Ax_n, \quad x_0 \text{ given}$$

Solution: $x_n = A^n x_0$
\end{block}

\begin{alertblock}{Key Insight}
Stability determined by eigenvalues of $A$:
\begin{itemize}
\item Continuous: $\text{Re}(\lambda_i) < 0$ for all $i$
\item Discrete: $|\lambda_i| < 1$ for all $i$
\end{itemize}
\end{alertblock}
\end{frame}

\begin{frame}{Eigenvalue Classification}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Eigenvalues & Continuous & Discrete \\
\hline
All $\text{Re}(\lambda) < 0$ & Stable node/spiral & \\
$|\lambda| < 1$ & & Stable node/spiral \\
\hline
Some $\text{Re}(\lambda) > 0$ & Unstable & \\
Some $|\lambda| > 1$ & & Unstable \\
\hline
Mixed signs & Saddle point & Saddle point \\
\hline
Pure imaginary & Center & \\
$|\lambda| = 1$ & & Neutral \\
\hline
\end{tabular}
\end{center}
\end{frame}

\section{Next Lecture Preview}

\begin{frame}{Coming Up Next}
\begin{block}{Lecture 2: Nonlinear Systems and Linearization}
\begin{itemize}
\item Hartman-Grobman theorem
\item Local stability analysis
\item Center manifold theory
\item Normal forms
\end{itemize}
\end{block}

\begin{block}{Reading Assignment}
\begin{itemize}
\item Strogatz, Chapter 2-3
\item Wiggins, Chapter 1
\end{itemize}
\end{block}
\end{frame}

\end{document}
```

## Lecture 2: Nonlinear Systems and Linearization

```latex
\documentclass{beamer}
\usetheme{Madrid}
\usecolortheme{default}

\usepackage{amsmath,amssymb,amsfonts}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{pgfplots}

\title[Dynamical Systems]{Nonlinear Systems and Linearization}
\subtitle{Lecture 2: Local Analysis and Stability}
\author{Course Instructor}
\institute{University Name}
\date{\today}

\begin{document}

\frame{\titlepage}

\section{Linearization of Nonlinear Systems}

\begin{frame}{Motivation for Linearization}
\begin{block}{The Challenge}
Nonlinear systems: $\dot{x} = f(x)$ where $f$ is nonlinear
\begin{itemize}
\item Generally no closed-form solutions
\item Behavior can be extremely complex
\item Need local analysis techniques
\end{itemize}
\end{block}

\begin{block}{The Strategy}
Near equilibrium $x^*$: approximate by linear system
$$f(x) \approx f(x^*) + Df(x^*)(x - x^*)$$

Since $f(x^*) = 0$:
$$\dot{x} \approx Df(x^*)(x - x^*)$$
\end{block}
\end{frame}

\begin{frame}{Jacobian Matrix}
\begin{definition}[Jacobian]
For $f: \mathbb{R}^n \to \mathbb{R}^n$:
$$J = Df(x^*) = \begin{pmatrix}
\frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\
\vdots & \ddots & \vdots \\
\frac{\partial f_n}{\partial x_1} & \cdots & \frac{\partial f_n}{\partial x_n}
\end{pmatrix}_{x=x^*}$$
\end{definition}

\begin{example}
Van der Pol oscillator: $\ddot{x} - \mu(1-x^2)\dot{x} + x = 0$

Convert: $\dot{x} = y$, $\dot{y} = \mu(1-x^2)y - x$

At $(0,0)$: $J = \begin{pmatrix} 0 & 1 \\ -1 & \mu \end{pmatrix}$
\end{example}
\end{frame}

\section{Hartman-Grobman Theorem}

\begin{frame}{The Fundamental Result}
\begin{theorem}[Hartman-Grobman]
Let $x^*$ be a hyperbolic equilibrium of $\dot{x} = f(x)$ (i.e., no eigenvalues on imaginary axis). Then there exists a neighborhood $U$ of $x^*$ and a homeomorphism $h: U \to h(U)$ such that the flow of the nonlinear system in $U$ is topologically conjugate to the flow of the linear system $\dot{y} = Jy$.
\end{theorem}

\begin{block}{Implications}
\begin{itemize}
\item Local behavior near hyperbolic equilibria determined by linearization
\item Topology preserved (but not metric properties)
\item Stability/instability transfers from linear to nonlinear
\end{itemize}
\end{block}

\begin{alertblock}{Limitation}
Doesn't apply when eigenvalues have zero real parts!
\end{alertblock}
\end{frame}

\begin{frame}{Stability Classification}
\begin{center}
\begin{tikzpicture}[scale=0.8]
% Create a 2D phase portrait examples
\begin{scope}[shift={(0,0)}]
\node at (0,-1.5) {\textbf{Stable Node}};
\draw[->] (-1,-1) -- (0,0);
\draw[->] (1,-1) -- (0,0);
\draw[->] (-1,1) -- (0,0);
\draw[->] (1,1) -- (0,0);
\draw[->] (0,-1) -- (0,0);
\draw[->] (0,1) -- (0,0);
\end{scope}

\begin{scope}[shift={(3,0)}]
\node at (0,-1.5) {\textbf{Unstable Node}};
\draw[->] (0,0) -- (-1,-1);
\draw[->] (0,0) -- (1,-1);
\draw[->] (0,0) -- (-1,1);
\draw[->] (0,0) -- (1,1);
\draw[->] (0,0) -- (0,-1);
\draw[->] (0,0) -- (0,1);
\end{scope}

\begin{scope}[shift={(6,0)}]
\node at (0,-1.5) {\textbf{Saddle Point}};
\draw[->] (-1,0) -- (0,0);
\draw[->] (1,0) -- (0,0);
\draw[->] (0,0) -- (0,-1);
\draw[->] (0,0) -- (0,1);
\end{scope}
\end{tikzpicture}
\end{center}

\begin{align}
\text{Stable: } &\text{Re}(\lambda_i) < 0 \text{ for all } i \\
\text{Unstable: } &\text{Re}(\lambda_i) > 0 \text{ for some } i \\
\text{Saddle: } &\text{Mixed signs in Re}(\lambda_i)
\end{align}
\end{frame}

\section{Center Manifold Theory}

\begin{frame}{When Linearization Fails}
\begin{problem}
What happens when some eigenvalues have zero real part?
\end{problem}

\begin{example}
$$\dot{x} = -x + xy, \quad \dot{y} = -y^2$$

Jacobian at $(0,0)$: $J = \begin{pmatrix} -1 & 0 \\ 0 & 0 \end{pmatrix}$

Eigenvalues: $\lambda_1 = -1$, $\lambda_2 = 0$
\end{example}

\begin{block}{The Issue}
\begin{itemize}
\item Linearization gives neutral stability in $y$ direction
\item Nonlinear terms determine actual behavior
\item Need center manifold theory
\end{itemize}
\end{block}
\end{frame}

\begin{frame}{Center Manifold Theorem}
\begin{theorem}[Center Manifold]
Consider $\dot{x} = Ax + f(x,y)$, $\dot{y} = By + g(x,y)$ where:
\begin{itemize}
\item $A$ has eigenvalues with $\text{Re}(\lambda) < 0$
\item $B$ has eigenvalues with $\text{Re}(\lambda) = 0$
\item $f(0,0) = g(0,0) = 0$, $Df(0,0) = Dg(0,0) = 0$
\end{itemize}

Then there exists a center manifold $W^c = \{(x,y) : x = h(y), |y| < \delta\}$ where $h(0) = 0$, $Dh(0) = 0$.
\end{theorem}

\begin{block}{Key Point}
Stability determined by reduced system on center manifold:
$$\dot{y} = By + g(h(y), y)$$
\end{block}
\end{frame}

\begin{frame}{Center Manifold Example}
\begin{example}
$$\dot{x} = -x + xy, \quad \dot{y} = -y^2$$

For center manifold $x = h(y)$:
$$\dot{x} = \frac{dh}{dy}\dot{y} = -x + xy$$

So: $\frac{dh}{dy}(-y^2) = -h(y) + h(y)y$

This gives: $h(y) = y + O(y^2)$

Reduced dynamics: $\dot{y} = -y^2$

Therefore: origin is asymptotically stable!
\end{example}

\begin{alertblock}{Caution}
Center manifold is not unique, but dynamics on it determine stability.
\end{alertblock}
\end{frame}

\section{Normal Forms}

\begin{frame}{Normal Form Theory}
\begin{block}{Motivation}
Simplify systems near equilibria by eliminating "non-essential" nonlinear terms through coordinate changes.
\end{block}

\begin{definition}[Normal Form]
A dynamical system is in normal form if it cannot be simplified further by polynomial coordinate changes.
\end{definition}

\begin{example}[Generic Pitchfork Bifurcation]
$$\dot{x} = \mu x - x^3$$
This is already in normal form.
\end{example}

\begin{example}[Before Normal Form]
$$\dot{x} = \mu x - x^3 + x^2y + xy^2$$
$$\dot{y} = -y + x^2$$
Can be simplified using center manifold reduction.
\end{example}
\end{frame}

\begin{frame}{Normal Form Procedure}
\begin{enumerate}
\item Move equilibrium to origin
\item Linearize and find Jordan normal form
\item Apply successive coordinate transformations to eliminate non-resonant terms
\item Keep only resonant terms (determined by eigenvalue relationships)
\end{enumerate}

\begin{block}{Resonance Conditions}
For eigenvalues $\lambda_1, \ldots, \lambda_n$ and multi-index $k = (k_1, \ldots, k_n)$:

Term $x^k$ is resonant if $\lambda_j = k \cdot \lambda$ for some $j$, where $k \cdot \lambda = \sum k_i \lambda_i$.
\end{block}

\begin{alertblock}{Complexity}
Normal forms become increasingly complex in higher dimensions and higher-order terms.
\end{alertblock}
\end{frame}

\section{Lyapunov's Direct Method}

\begin{frame}{Lyapunov Functions}
\begin{definition}[Lyapunov Function]
For system $\dot{x} = f(x)$ with equilibrium at origin, $V(x)$ is a Lyapunov function if:
\begin{enumerate}
\item $V(0) = 0$
\item $V(x) > 0$ for $x \neq 0$ in some neighborhood of 0
\item $\dot{V}(x) = \nabla V \cdot f(x) \leq 0$ along trajectories
\end{enumerate}
\end{definition}

\begin{theorem}[Lyapunov Stability]
\begin{itemize}
\item If $\dot{V} \leq 0$: equilibrium is stable
\item If $\dot{V} < 0$ for $x \neq 0$: equilibrium is asymptotically stable
\end{itemize}
\end{theorem}

\begin{alertblock}{ converse}
If equilibrium is asymptotically stable, then a Lyapunov function exists (but finding it may be difficult).
\end{alertblock}
\end{frame}

\begin{frame}{Lyapunov Function Examples}
\begin{example}[Mechanical System]
$$\dot{x} = y, \quad \dot{y} = -x - \gamma y$$

Energy function: $V(x,y) = \frac{1}{2}(x^2 + y^2)$

$$\dot{V} = x\dot{x} + y\dot{y} = xy + y(-x - \gamma y) = -\gamma y^2 \leq 0$$

If $\gamma > 0$: asymptotically stable.
\end{example}

\begin{example}[Nonlinear System]
$$\dot{x} = -x + 2xy, \quad \dot{y} = -y$$

Try: $V(x,y) = x^2 + y^2$

$$\dot{V} = 2x(-x + 2xy) + 2y(-y) = -2x^2 + 4x^2y - 2y^2$$

This doesn't work globally, but works locally near origin.
\end{example}
\end{frame}

\section{Applications to Different Fields}

\begin{frame}{Control Theory Application}
\begin{example}[Feedback Linearization]
Consider: $\dot{x} = f(x) + g(x)u$

Goal: Find control $u$ to stabilize equilibrium.

1. Linearize: $\dot{x} = Ax + bu$ near equilibrium
2. Design linear controller: $u = -kx$
3. Analyze nonlinear closed-loop stability using Lyapunov methods
\end{example}

\begin{block}{LaSalle's Invariance Principle}
If $\dot{V} \leq 0$ and trajectories are bounded, then trajectories approach the largest invariant set where $\dot{V} = 0$.

Crucial for proving global stability.
\end{block}
\end{frame}

\begin{frame}{Neural Network Dynamics}
\begin{example}[Hopfield Network Stability]
$$\tau_i \dot{x_i} = -x_i + \sum_j w_{ij} \sigma(x_j) + I_i$$

Lyapunov function (when $w_{ij} = w_{ji}$):
$$V = -\frac{1}{2}\sum_{ij} w_{ij}\sigma(x_i)\sigma(x_j) - \sum_i I_i\sigma(x_i) + \sum_i \int_0^{x_i} \sigma^{-1}(s)ds$$

$$\dot{V} = -\sum_i \frac{1}{\tau_i}(\dot{x_i})^2 \leq 0$$

Network converges to fixed points (acts as associative memory).
\end{example}
\end{frame}

\section{Next Lecture Preview}

\begin{frame}{Coming Up Next}
\begin{block}{Lecture 3: Bifurcation Theory}
\begin{itemize}
\item Saddle-node, transcritical, pitchfork bifurcations
\item Hopf bifurcations and limit cycles
\item Codimension-2 bifurcations
\item Applications to pattern formation
\end{itemize}
\end{block}

\begin{block}{Homework Assignment}
\begin{itemize}
\item Analyze stability of given nonlinear systems
\item Compute center manifolds
\item Find Lyapunov functions
\end{itemize}
\end{block}
\end{frame}

\end{document}
```

## Lecture 3: Bifurcation Theory

```latex
\documentclass{beamer}
\usetheme{Madrid}
\usecolortheme{default}

\usepackage{amsmath,amssymb,amsfonts}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{pgfplots}

\title[Dynamical Systems]{Bifurcation Theory}
\subtitle{Lecture 3: Local Bifurcations and Pattern Formation}
\author{Course Instructor}
\institute{University Name}
\date{\today}

\begin{document}

\frame{\titlepage}

\section{Introduction to Bifurcations}

\begin{frame}{What is a Bifurcation?}
\begin{definition}[Bifurcation]
A qualitative change in the dynamics of a system as parameters vary, typically involving:
\begin{itemize}
\item Creation/destruction of equilibria
\item Changes in stability
\item Birth/death of periodic orbits
\item Changes in topology of phase space
\end{itemize}
\end{definition}

\begin{block}{Mathematical Setup}
Consider parameterized family: $\dot{x} = f(x, \mu)$ where $\mu \in \mathbb{R}^k$

\textbf{Bifurcation point}: Parameter value $\mu_0$ where qualitative behavior changes.
\end{block}

\begin{example}
$\dot{x} = \mu - x^2$
\begin{itemize}
\item $\mu > 0$: two equilibria
\item $\mu = 0$: one equilibrium  
\item $\mu < 0$: no equilibria
\end{itemize}
\end{example}
\end{frame}

\begin{frame}{Bifurcation Conditions}
\begin{theorem}[Generic Bifurcation Condition]
For $\dot{x} = f(x,\mu)$, bifurcations typically occur when:
\begin{enumerate}
\item At equilibrium: $f(x^*,\mu_0) = 0$
\item Jacobian has eigenvalue with zero real part: $\text{Re}(\lambda) = 0$
\item Non-degeneracy conditions on higher derivatives
\end{enumerate}
\end{theorem}

\begin{block}{Codimension}
\begin{itemize}
\item \textbf{Codimension-1}: Generic in 1-parameter families
\item \textbf{Codimension-2}: Require 2 parameters to observe
\item Higher codimension: Increasingly special
\end{itemize}
\end{block}

\begin{alertblock}{Structural Stability}
Most systems are structurally stable: small perturbations don't change qualitative behavior. Bifurcations occur at boundaries between different stable regions.
\end{alertblock}
\end{frame}

\section{Codimension-1 Bifurcations}

\begin{frame}{Saddle-Node Bifurcation}
\begin{block}{Normal Form}
$$\dot{x} = \mu + x^2$$
\end{block}

\begin{columns}
\column{0.5\textwidth}
\begin{block}{Analysis}
\begin{itemize}
\item $\mu < 0$: two equilibria
  \begin{align}
  x^*_{\pm} = \pm\sqrt{-\mu}
  \end{align}
\item $x^*_-$: stable
\item $x^*_+$: unstable  
\item $\mu = 0$: saddle-node
\item $\mu > 0$: no equilibria
\end{itemize}
\end{block}

\column{0.5\textwidth}
\begin{center}
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
    xlabel = $\mu$,
    ylabel = $x$,
    xmin=-1, xmax=1,
    ymin=-2, ymax=2,
    axis lines = center,
]
\addplot[blue, thick, domain=-1:0] {sqrt(-x)};
\addplot[red, dashed, thick, domain=-1:0] {-sqrt(-x)};
\node at (axis cs: -0.5, 1) {\small unstable};
\node at (axis cs: -0.5, -1) {\small stable};
\end{axis}
\end{tikzpicture}
\end{center}
\end{columns}

\begin{alertblock}{Scaling}
Near bifurcation: $x \sim \mu^{1/2}$, $t \sim \mu^{-1/2}$
\end{alertblock}
\end{frame}

\begin{frame}{Transcritical Bifurcation}
\begin{block}{Normal Form}
$$\dot{x} = \mu x - x^2$$
\end{block}

\begin{columns}
\column{0.5\textwidth}
\begin{block}{Analysis}
Equilibria: $x^* = 0, \mu$

Stability:
\begin{itemize}
\item $\mu < 0$: $x=0$ stable, $x=\mu$ unstable
\item $\mu > 0$: $x=0$ unstable, $x=\mu$ stable
\end{itemize}

\textbf{Exchange of stability}
\end{block}

\column{0.5\textwidth}
\begin{center}
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
    xlabel = $\mu$,
    ylabel = $x$,
    xmin=-1, xmax=1,
    ymin=-1, ymax=1,
    axis lines = center,
]
\addplot[blue, thick, domain=-1:0] {0};
\addplot[red, dashed, thick, domain=0:1] {0};
\addplot[red, dashed, thick, domain=-1:0] {x};
\addplot[blue, thick, domain=0:1] {x};
\end{axis}
\end{tikzpicture}
\end{center}
\end{columns}

\begin{example}[Logistic Growth]
$\dot{N} = rN(1 - N/K)$ with $r$ as bifurcation parameter.
\end{example}
\end{frame}

\begin{frame}{Pitchfork Bifurcation}
\begin{columns}
\column{0.5\textwidth}
\begin{block}{Supercritical}
$$\dot{x} = \mu x - x^3$$

\begin{itemize}
\item $\mu < 0$: $x=0$ stable
\item $\mu > 0$: $x=0$ unstable, $x=\pm\sqrt{\mu}$ stable
\end{itemize}
\end{block}

\column{0.5\textwidth}
\begin{block}{Subcritical}
$$\dot{x} = \mu x + x^3$$

\begin{itemize}
\item $\mu < 0$: $x=0$ stable, $x=\pm\sqrt{-\mu}$ unstable
\item $\mu > 0$: $x=0$ unstable
\end{itemize}
\end{block}
\end{columns}

\vspace{0.5cm}
\begin{center}
\begin{tikzpicture}[scale=0.7]
% Supercritical
\begin{scope}[shift={(-3,0)}]
\begin{axis}[
    xlabel = $\mu$,
    ylabel = $x$,
    xmin=-0.5, xmax=1,
    ymin=-1.5, ymax=1.5,
    axis lines = center,
    title = Supercritical,
    width=4cm, height=4cm
]
\addplot[blue, thick, domain=-0.5:0] {0};
\addplot[red, dashed, thick, domain=0:1] {0};
\addplot[blue, thick, domain=0:1] {sqrt(x)};
\addplot[blue, thick, domain=0:1] {-sqrt(x)};
\end{axis}
\end{scope}

% Subcritical  
\begin{scope}[shift={(3,0)}]
\begin{axis}[
    xlabel = $\mu$,
    ylabel = $x$,
    xmin=-1, xmax=0.5,
    ymin=-1.5, ymax=1.5,
    axis lines = center,
    title = Subcritical,
    width=4cm, height=4cm
]
\addplot[blue, thick, domain=-1:0] {0};
\addplot[red, dashed, thick, domain=0:0.5] {0};
\addplot[red, dashed, thick, domain=-1:0] {sqrt(-x)};
\addplot[red, dashed, thick, domain=-1:0] {-sqrt(-x)};
\end{axis}
\end{scope}
\end{tikzpicture}
\end{center}
\end{frame}

\section{Hopf Bifurcations}

\begin{frame}{Hopf Bifurcation Theory}
\begin{block}{Setup}
Consider system $\dot{x} = f(x,\mu)$ where at $\mu = 0$:
\begin{itemize}
\item $(x^*, \mu) = (0, 0)$ is equilibrium
\item Eigenvalues: $\lambda(\mu) = \alpha(\mu) \pm i\beta(\mu)$
\item At bifurcation: $\alpha(0) = 0$, $\beta(0) = \omega_0 \neq 0$
\item Transversality: $\alpha'(0) \neq 0$
\end{itemize}
\end{block}

\begin{theorem}[Hopf Bifurcation]
Under non-degeneracy conditions, there exists a branch of periodic orbits bifurcating from the equilibrium.

\textbf{Supercritical}: Stable limit cycles for $\mu > 0$
\textbf{Subcritical}: Unstable limit cycles for $\mu < 0$
\end{theorem}

\begin{alertblock}{Lyapunov Coefficient}
Sign of first Lyapunov coefficient $l_1$ determines criticality:
$l_1 < 0$ (supercritical), $l_1 > 0$ (subcritical)
\end{alertblock}
\end{frame}

\begin{frame}{Hopf Normal Form}
\begin{block}{Complex Normal Form}
$$\dot{z} = (\mu + i\omega_0)z + c z|z|^2 + O(|z|^4)$$
where $z = x + iy$ and $c = c_1 + ic_2$.
\end{block}

\begin{block}{Polar Coordinates}
$z = re^{i\theta}$ gives:
\begin{align}
\dot{r} &= \mu r + c_1 r^3 + O(r^5) \\
\dot{\theta} &= \omega_0 + c_2 r^2 + O(r^4)
\end{align}
\end{block}

\begin{itemize}
\item \textbf{Supercritical} ($c_1 < 0$): Stable limit cycle with $r \approx \sqrt{\mu/|c_1|}$
\item \textbf{Subcritical} ($c_1 > 0$): Unstable limit cycle
\item Frequency shift: $\omega \approx \omega_0 + c_2\mu$
\end{itemize}
\end{frame}

\begin{frame}{Hopf Bifurcation Examples}
\begin{example}[Van der Pol Oscillator]
$$\ddot{x} - \mu(1-x^2)\dot{x} + x = 0$$

Convert to first order:
\begin{align}
\dot{x} &= y \\
\dot{y} &= \mu(1-x^2)y - x
\end{align}

At $\mu = 0$: Hopf bifurcation at origin.
\begin{itemize}
\item $\mu < 0$: Stable spiral
\item $\mu > 0$: Supercritical Hopf $\to$ limit cycle
\end{itemize}
\end{example}

\begin{example}[Brusselator]
Chemical reaction model with limit cycle oscillations.
\end{example}
\end{frame}

\section{Codimension-2 Bifurcations}

\begin{frame}{Cusp Bifurcation}
\begin{block}{Normal Form}
$$\dot{x} = \mu_1 + \mu_2 x + x^3$$
\end{block}

\begin{columns}
\column{0.5\textwidth}
\begin{block}{Bifurcation Set}
In $(\mu_1, \mu_2)$ plane:
$$\mu_1 = \pm\frac{2}{3\sqrt{3}}|\mu_2|^{3/2}$$

Cusp point at origin.
\end{block}

\column{0.5\textwidth}
\begin{center}
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
    xlabel = $\mu_2$,
    ylabel = $\mu_1$,
    xmin=-1, xmax=1,
    ymin=-0.5, ymax=0.5,
    axis lines = center,
]
\addplot[blue, thick, domain=-1:0] {(2/3)*sqrt(3)*abs(x)^(1.5)};
\addplot[blue, thick, domain=-1:0] {-(2/3)*sqrt(3)*abs(x)^(1.5)};
\addplot[blue, thick, domain=0:1] {(2/3)*sqrt(3)*x^(1.5)};
\addplot[blue, thick, domain=0:1] {-(2/3)*sqrt(3)*x^(1.5)};

\node at (axis cs: 0.3, 0.3) {\small 1 eq.};
\node at (axis cs: -0.3, 0) {\small 3 eq.};
\node at (axis cs: 0.3, -0.3) {\small 1 eq.};
\end{axis}
\end{tikzpicture}
\end{center}
\end{columns}

Organizes saddle-node bifurcations in 2-parameter families.
\end{frame}

\begin{frame}{Bogdanov-Takens Bifurcation}
\begin{block}{Normal Form}
\begin{align}
\dot{x} &= y \\
\dot{y} &= \mu_1 + \mu_2 x + x^2 + xy
\end{align}
\end{block}

\begin{block}{Rich Dynamics}
Contains:
\begin{itemize}
\item Saddle-node bifurcations
\item Hopf bifurcation
\item Homoclinic bifurcation
\item All organized around codimension-2 point
\end{itemize}
\end{block}

\begin{alertblock}{Universal Unfolding}
The Bogdanov-Takens normal form is the universal unfolding of a equilibrium with double zero eigenvalue.
\end{alertblock}
\end{frame}

\section{Applications}

\begin{frame}{Pattern Formation in Biology}
\begin{example}[Reaction-Diffusion Systems]
$$\frac{\partial u}{\partial t} = f(u,v) + D_u \nabla^2 u$$
$$\frac{\partial v}{\partial t} = g(u,v) + D_v \nabla^2 v$$

\textbf{Turing instability}: Stable equilibrium becomes unstable to spatial perturbations when diffusion added.

Condition: $D_v/D_u$ large enough to destabilize uniform state.
\end{example}

\begin{block}{Pattern Selection}
\begin{itemize}
\item Wavelength selection through linear stability analysis
\item Weakly nonlinear analysis determines pattern amplitude
\item Secondary bifurcations create pattern transitions
\end{itemize}
\end{block}
\end{frame}

\begin{frame}{Engineering Applications}
\begin{example}[Control System Design]
Hopf bifurcations in feedback systems:
$$\dot{x} = f(x) + g(x)u, \quad u = k(x)$$

Design $k(x)$ to:
\begin{itemize}
\item Avoid Hopf bifurcations (no oscillations)
\item Or induce desired oscillatory behavior
\end{itemize}
\end{example}

\begin{example}[Power Systems]
Voltage collapse as saddle-node bifurcation:
$$\dot{V} = P_{\text{gen}}(V) - P_{\text{load}}(V)$$

Critical loading corresponds to saddle-node bifurcation point.
\end{example}
\end{frame}

\begin{frame}{Neural Networks and Oscillations}
\begin{example}[Wilson-Cowan Model]
$$\tau_E \dot{E} = -E + f(w_{EE}E - w_{EI}I + h_E)$$
$$\tau_I \dot{I} = -I + f(w_{IE}E - w_{II}I + h_I)$$

Bifurcations control:
\begin{itemize}
\item Transition from rest to active state
\item Emergence of oscillatory activity
\item Synchronization phenomena
\end{itemize}
\end{example}

\begin{alertblock}{Brain Dynamics}
Many neurological conditions associated with bifurcations:
\begin{itemize}
\item Epilepsy: transition to high-frequency oscillations
\item Parkinson's: emergence of pathological rhythms
\end{itemize}
\end{alertblock}
\end{frame}

\section{Computational Methods}

\begin{frame}{Bifurcation Analysis Software}
\begin{block}{Numerical Continuation}
\begin{itemize}
\item \textbf{AUTO}: Continuation of equilibria, periodic orbits
\item \textbf{MATCONT}: MATLAB-based bifurcation analysis
\item \textbf{PyDSTool}: Python dynamical systems toolbox
\end{itemize}
\end{block}

\begin{block}{Key Algorithms}
\begin{itemize}
\item Pseudo-arclength continuation
\item Detection of bifurcation points
\item Computation of normal forms
\item Branch switching at bifurcation points
\end{itemize}
\end{block}

\begin{example}[Typical Workflow]
\begin{enumerate}
\item Find equilibrium for given parameters
\item Continue equilibrium as parameter varies
\item Detect bifurcation points automatically
\item Switch to bifurcating branches (e.g., periodic orbits)
\item Compute stability and normal forms
\end{enumerate}
\end{example}
\end{frame}

\section{Next Lecture Preview}

\begin{frame}{Coming Up Next}
\begin{block}{Lecture 4: Chaos and Strange Attractors}
\begin{itemize}
\item Definition and characterization of chaos
\item Lyapunov exponents
\item Strange attractors and fractal dimensions
\item Routes to chaos
\item Applications in various fields
\end{itemize}
\end{block}

\begin{block}{Reading Assignment}
\begin{itemize}
\item Strogatz, Chapters 8-9
\item Kuznetsov, Chapter 5
\end{itemize}
\end{block}
\end{frame}

\end{document}
```

This comprehensive lecture series covers the fundamentals of dynamical systems theory appropriate for PhD students across multiple disciplines. Each lecture builds upon previous material while introducing new concepts with both theoretical rigor and practical applications.

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