These notes are taken by a student enrolled in the course. They are not edited and I am providing them in order to give you an idea of what is going on in the class.
Lecture 1: Course mechanics; Introduction to nonlinear systems
Lecture 2: Range of nonlinear phenomena: finite escape time, multiple isolated equilibria
Lecture 3: Limit cycles, chaos; Fold bifurcation
Lecture 4: Transcritical and pitchfork bifurcations; Phase portraits of 2nd order linear systems
Lecture 5: Phase portraits of nonlinear systems near hyperbolic equilibria (Hartman-Grobman theorem)
Lecture 6: Bendixon's theorem; Invariant sets
Lecture 7: Poincare-Bendixon theorem; Hopf bifurcations
Lecture 8: Hopf bifurcations (continued); Dimensional analysis and scaling
Lecture 9: Center manifold theory
Lecture 10: Existence and uniqueness of solutions
Lecture 11: Lipschitz continuity; Continuous dependence on initial conditions and parameters
Lecture 12: Effect of parameter variations on solutions; Sensitivity equations
Lecture 13: Lyapunov stability; Lyapunov's direct method
Lecture 14: Lyapunov functions (examples)
Lecture 15: LaSalle's invariance principle
Lecture 16: Lyapunov functions for linear systems
Lecture 17: Stability via linearization; Integrator backstepping
Lecture 18: Integrator backstepping (continued)
Lecture 19: Control Lyapunov functions; Sontag's formula
Lecture 20: Comparison functions; Stability of time-varying systems
Lecture 21: Lyapunov functions for time-varying systems
Lecture 22: Linear time-varying systems; Differential Lyapunov Equation