Nonlinear Dynamics and Chaos

Catalog description

Continuous Dynamical Systems: Nonlinear Equations versus Linear Equations, Flows on a Line, Fixed Points and Stability, Stability Analysis, Potentials, Saddle-Node Bifurcations, Transcritical Bifurcations, Supercritical and Subcritical Pitchfork Bifurcations, Dimensional Analysis and Scaling, Imperfect Bifurcations and Catastrophes, Flows on the Circle, The Uniform and Nonuniform Oscillator, Oscillation Periods, Two-Dimensional Flows, Linear Systems, Eigenvalues and Eigenvectors, Classification of Fixed Points, Phase Portraits, Existence and Uniqueness, Fixed Points and Linearization, Nonlinear Terms, Conservative Systems, Reversible Systems, Index Theory, Limit Cycles, Van Der Pol Oscillator, Gradient Systems, Liaponov Functions, Dulac's Criterion, Poincare-Bendixson Theorem, Lienard Systems, Relaxation Oscillations, Weakly Nonlinear Oscillators, Regular Perturbation Theory, Two-Timing, Supercritical and Subcritical Hopf Bifurcations, Global Bifurcations of Cycles, Hysteresis, and the Poincare Map, The Lorenz Equations, Strange Attractors, The Lorenz Map. Prerequisite: Some Differential Equations and Senior or Master's standing in Engineering or permission of the instructor. Discrete Dynamical Systems: One-Dimensional Maps, Fixed Points and Cobwebs, The Logistic Map, Periodic Windows, Period Doubling, The Liapunov Exponent, Universality, Feigenbaum's Number, Feigenbaum's Renormalization Theory, Fractals, Countable and Uncountable Sets, The Cantor Set, Self-Similar Fractals and Their Dimensions, The von Koch Curve, Box Dimension, Multifractals.