Nonlinear Dynamical Systems

Overview

Introduction - First Order systems - Classification of Equilibrium points - Lipschitz Functions - Existence uniqueness theorems - Existence uniqueness of solutions to differential equations - Lyapunov theorem on stability - Extension of Lyapunov's Theorem in different contexts - LaSalle's Invariance principle, Barbashin and Krasovski theorems, periodic orbits - Bendixson criterion and Poincare-Bendixson criterion.Ex:Lotka Volterra predator prey - Bendixson and Poincare-Bendixson criteria van-der-Pol Oscillator - Scilab simulation of Lotka Volterra predator prey model, van-der-Pol Oscillator - Signals, operators - Norms of signals, systems (operators), Finite gain L2 stable - Nyquist plots and Nyquist criterion for stability - Interconnection between linear system & non-linearity, passive filters - Passive filters, Dissipation equality, positive real lemma - Positive real lemma proof - Definition for positive realness and Kalman Yakubovich-Popov Theorem - Kalman-Yakubovich-Popov Lemma/theorem and memoryless nonlinearities

Loop tranformations and circle criterion - Nonlinearities based on circle criterion - Limit cycles - Popov criterion continuous, frequency-domain theorem - Describing function method - Describing Function - Describing : optimal gain - Describing function : Jump Hysteresis - Describing functions:sufficient conditions for existence of periodic orbits - Describing functions for nonlinearities - Ideal relay with Hysteresis and dead zone - Dynamical systems on manifolds