INTRODUCTION TO DYNAMICAL SYSTEMS
Course Notes for AM 451

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Contents

• Chapter 1: Introduction
  • 1.1 Qualitative analysis
  • 1.2 Definitions and notations
  • 1.3 Equilibrium points and linearization
• Chapter 2: Linear Systems
  • 2.1 Linear systems in R²
  • 2.2 Exponential of matrices
  • 2.3 Distinct eigenvalues
  • 2.4 Multiple eigenvalues
  • 2.5 Asymptotic behaviour
  • 2.6 Contractions and expansions
• Chapter 3: Nonlinear Systems
  • 3.1 Nonlinear sinks and sources
  • 3.2 Fundamental theory
  • 3.3 The flow defined by ODEs
  • 3.4 The Hartman-Grobman theorem
  • 3.5 Stability of equilibria
  • 3.6 Hamiltonian systems
  • 3.7 Limit sets and long-term behaviour
• Chapter 4: Periodic Solutions and Bifurcations of Vector Fields
  • 4.1 Limit cycles and separatrix cycles
  • 4.2 Dulac's criteria
  • 4.3 Poincaré-Bendixson theorem
  • 4.4 Lienard systems
  • 4.5 Structural stability and bifurcation
  • 4.6 Hopf bifurcations
• Appendix A: Maple Worksheet
  • A.1 Maple worksheet for chapter 2
  • A.2 Maple worksheet for chapter 3
  • A.3 Maple worksheet for chapter 4
• Appendix B: ODE Phase Portraits Plotter

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