CONTENTS: Linear and nonlinear systems of ordinary differential equations; geometric approach to dynamical systems (vector fields and iterated maps); bifurcations; aspects of chaotic dynamics; physical, chemical and biological models. I. 1-Dimensional Flows: Basic concepts, linear vs. nonlinear systems, linearization. Steady state bifurcations: saddle node, transcritical, pitchfork and imperfect bifurcations. Flows on circles and infinite period bifurcations. II. 2-Dimensional Flows: Classification of linear systems, phase space analysis, conservative and reversible systems. Limit cycles, relaxation oscillators vs. weakly nonlinear oscillators. Steady state bifurcations in 2-d, Hopf bifurcation. Coupled oscillators, Poincare maps. III. Chaos: Lorenz equations, Lorenz map and its dependence on parameters. 1-dimensional maps: Liapunov exponents, renormalization. Fractals: Cantor sets and fractal dimension. Examples of strange attractors.
Homework with optional final.
If a final is taken it counts 40%, otherwise the grading will be based purely on homework.