EE 587 – Lecture Notes

Mihailo Jovanovic, University of Southern California, Spring 2023

These notes are not edited but they should give you a fairly good idea of what is going on during lectures.

  1. Course mechanics; Topics; Introduction to nonlinear systems; Examples of nonlinear systems

  2. Equilibrium points; Fluctuations’ dynamics; Linearization; Flows of first-order systems (i.e., scalar ODEs)

  3. Local analysis via linearization; Range of phenomena for nonlinear systems (multiple isolated equilibrium points)

  4. Range of phenomena for nonlinear systems (finite escape time, limit cycles)

  5. Range of phenomena for nonlinear systems (chaos); Bifurcations in first-order systems (fold, transcritical)

  6. Bifurcations in first-order systems (pitchfork); Phase portraits of 2nd order linear time-invariant systems

  7. Phase portraits of nonlinear systems near hyperbolic equilibria (Hartman-Grobman theorem)

  8. Bendixon's theorem; Positively invariant sets

  9. Positively invariant sets (examples); Poincare-Bendixon's theorem

  10. Hopf bifurcations

  11. Center manifold theory

  12. Center manifold theory (continued)

  13. Existence and uniqueness of solutions; Lipschitz continuity; Continuous dependence on initial conditions and parameters

  14. Sensitivity equations; Stability of equilibrium points

  15. Lyapunov functions; Positive definiteness; Radial unboundedness

  16. Lyapunov-based stability analysis

  17. Lyapunov functions (examples); LaSalle's invariance principle; Lyapunov functions for LTI systems

  18. Algebraic Lyapunov equation; LaSalle's invariance principle for LTI systems