Introduction to Chaos Theory
- Free oscillators: Linear and nonlinear pendulum. Phase space and phase portraits. Fixed points. Stability. Liouville's theorem and conservation of areas in phase space. Damped pendulum: Attractors, Contraction of areas in phase space.
- Forced oscillators: Van der Pol equation. Limit cycles. Forced pendulum. Resonance. Stability. Introduction to bifurcation theory.
- Methods for analyzing periodic, quasiperiodic, and aperiodic systems. Poincare sections. Floquet matrices and stability. Maps. Reduction of ﬂows to maps. Reconstruction of phase space from one-dimensional signals.
- Strange attractors. Dissipation, attraction, and reduction of dimensionality. Minimum dimension of deterministic aperiodic systems. Sensitivity to initial conditions. Stretching and folding. Rossler attractor. Derivation of Lorenz attractor. Stability of Lorenz equations. Henon attractor.
- Quantitative analysis of strange attractors. Lyaponov exponents. Fractal dimension.
- Transitions to chaos. Period doubling: logistic map, Feigenbaum numbers, scaling, and universality. Quasiperiodicity. Intermittency. Illustrations from experiments.
- Various applications of the course material in geosciences.