Chaotic Dynamical Systems

This course introduces the ideas of determinism and randomness in the physical world.

* Introduction to phase plane, critical points and characterization (hyperbolic/elliptic) - free and damped oscillators, prey-predator models

* Simple extensions to three-dimensional phase space and beyond, e.g. rotation of rigid bodis, the Lorenz system

* Integrable and non-integrable systems, Poincare return maps

* Discrete dynamics - 1D and 2D maps; fixed points and stability; period doubling - shift map, logistic map

* Breakdown of order and chaos; sensitivity to initial conditions ("butterfly effect") and Lyapunov exponents; limit to predictability; strange attractors and fractal dimension - Kepler problem, Henon-Heiles system

* Stable and unstable manifolds, homoclinic and heteroclinic tangle, lobes and turnstile transport, particle motion in 2D incompressible fluid