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; Fold bifurcation
Lecture 3: Transcritical and pitchfork bifurcations; Phase portraits of 2nd order linear systems
Lecture 4: Phase portraits of nonlinear systems near hyperbolic equilibria (Hartman-Grobman); Bendixon's theorem
Lecture 5: Bendixon's theorem (examples); Invariant sets; Poincare-Bendixon theorem
Lecture 6: Hopf bifurcations (super- and sub-critical); Dimensional analysis and scaling
Lecture 7: Center manifold theory
Lecture 8: Existence and uniqueness of solutions; Lipschitz continuity
Lecture 9: Continuous dependence on initial conditions and parameters; Effect of parameter variations on solutions; Sensitivity equations
Lecture 10: Lyapunov stability; Lyapunov's direct method
Lecture 11: Lyapunov functions (examples); LaSalle's invariance principle
Lecture 12: Lyapunov functions for linear systems; Stability via linearization
Lecture 13: Comparison functions; Stability of time-varying systems
Lecture 14: Lyapunov functions for time-varying systems
Lecture 15: Linear time-varying systems; Differential Lyapunov Equation
Lecture 16: Uniform observability; Gradient algorithm for estimation of unknown parameters
Lecture 17: Convergence of gradient algorithm for estimation of unknown parameters
Lecture 18: Model reference adaptive control; Introduction to backstepping
Lecture 19: Integrator backstepping; Control Lyapunov functions
Lecture 20: Input-output stability
Lecture 21: Hamilton-Jacobi inequality; Bounded-real lemma; Small-gain theorem
Lecture 22: Passivity
Lecture 23: Positive real transfer functions; Kalman-Yakubovich-Popov Lemma; Passivity theorem
Lecture 24: Passivity theorem for large-scale interconnected systems
Lecture 25: Input-to-state stability; Relative degree
Lecture 26: Input-output linearization; Zero dynamics
Lecture 27: Normal form
Lecture 28: Course summary