Dynamical Systems Theory

 

• Introduction
• Flows and Maps
  • Flows
  • Poincaré Map
  • Suspension of a Map
  • Creed and Quest
• Asymptotic Behavior and Recurrence
  • Invariant Sets
  • Limit Sets: , , and Nonwandering
  • Chain Recurrence
• Expansions and Contractions
  • Derivative of a Map
  • Jacobian of a Map
  • Divergence of a Vector Field
  • Dissipative and Conservative
  • Equation of First Variation
• Fixed Points
  • Stability
  • Linearization
  • Hyperbolic Fixed Points: . Saddles, Sources, and Sinks
• Invariant Manifolds
  • Center Manifold Theorem
  • Homoclinic and Heteroclinic Points
• Example: Laser Equations
  • Steady States
  • Eigenvalues of a Matrix
  • Eigenvectors
  • Stable Focus
• Smale Horseshoe
  • From Tangles to Horseshoes
  • Horseshoe Map
  • Symbolic Dynamics
  • From Horseshoes to Tangles
• Hyperbolicity
• Lyapunov Characteristic Exponent