These notes are not edited but they should give you a fairly good idea of what is going on during lectures.
Course mechanics; Topics; Introduction to nonlinear systems; Examples of nonlinear systems
Equilibrium points; Fluctuations’ dynamics; Linearization; Flows of first-order systems (i.e., scalar ODEs)
Local analysis via linearization; Range of phenomena for nonlinear systems (multiple isolated equilibrium points)
Range of phenomena for nonlinear systems (finite escape time, limit cycles)
Range of phenomena for nonlinear systems (chaos); Bifurcations in first-order systems (fold, transcritical)
Bifurcations in first-order systems (pitchfork); Phase portraits of 2nd order linear time-invariant systems
Phase portraits of nonlinear systems near hyperbolic equilibria (Hartman-Grobman theorem)
Bendixon's theorem; Positively invariant sets
Positively invariant sets (examples); Poincare-Bendixon's theorem
Hopf bifurcations
Center manifold theory
Center manifold theory (continued)
Existence and uniqueness of solutions; Lipschitz continuity; Continuous dependence on initial conditions and parameters
Sensitivity equations; Stability of equilibrium points
Lyapunov functions; Positive definiteness; Radial unboundedness
Lyapunov-based stability analysis
Lyapunov functions (examples); LaSalle's invariance principle; Lyapunov functions for LTI systems
Algebraic Lyapunov equation; LaSalle's invariance principle for LTI systems