Flows on the line-A Geometric Way of Thinking, Fixed Points and Stability, Population Growth, Linear Stability Analysis, Existence and Uniqueness, Solving Equations on the Computer. Bifurcations-Saddle-Node Bifurcation, Transcritical Bifurcation, Pitchfork Bifurcation.
Linear Systems-Definitions and Examples, Classification of Linear Systems. Phase Plane- Phase Portraits, Fixed Points and Linearization, Index Theory. Limit Cycles- Ruling Out Closed Orbits, Poincare-Bendixson Theorem, Lienard Systems, Relaxation Oscillators, Weakly Nonlinear Oscillators. Bifurcations Saddle-Node, Transcritical, and Pitchfork Bifurcations, Hopf-Bifurcations. Oscillating Chemical Reactions, Global Bifurcations of Cycles, Hysteresis in the Driven Pendulum and Josephson Junction, Coupled Oscillators and Quasiperiodicity, Poincare Maps.
Lorenz Equations-Simple Properties of the Lorenz Equations, Chaos on a Strange Attractor, Lorenz Map, Exploring Parameter Space. One-Dimensional aps- Fixed Points and Cobwebs, Logistic Map: Logistic Map: Analysis, Periodic Windows, Liapunov Exponent, Universality and Renormalization.