Nonlinear Dynamical Systems


Lecture 1 - Introduction


Lecture 2 - First Order systems


Lecture 3 - Classification of Equilibrium points


Lecture 4 - Lipschitz Functions


Lecture 5 - Existence/uniqueness theorems


Lecture 6 - Existence/uniqueness of solutions to differential equations


Lecture 7 - Lyapunov theorem on stability


Lecture 8 - Extension of Lyapunov's Theorem in different contexts


Lecture 9 - LaSalle's Invariance principle, Barbashin and Krasovski theorems, periodic orbits


Lecture 10 - Bendixson criterion and Poincare-Bendixson criterion. Example: Lotka Volterra predator prey model


Lecture 11 - Bendixson and Poincare-Bendixson criteria van-der-Pol Oscillator


Lecture 12 - Scilab simulation of Lotka Volterra predator prey model, van-der-Pol Oscillator Review of linearization, operating point/operating trajectory


Lecture 13 - Signals, operators


Lecture 14 - Norms of signals, systems (operators), Finite gain L2 stable


Lecture 15 - Nyquist plots and Nyquist criterion for stability


Lecture 16 - Interconnection between linear system & non-linearity, passive filters


Lecture 17 - Passive filters, Dissipation equality, positive real lemma


Lecture 18 - Positive real lemma proof


Lecture 19 - Definition for positive realness and Kalman Yakubovich-Popov Theorem


Lecture 20 - Kalman-Yakubovich-Popov Lemma/theorem and memoryless nonlinearities


Lecture 21 - Loop tranformations and circle criterion


Lecture 22 - Nonlinearities based on circle criterion


Lecture 23 - Limit cycles


Lecture 24 - Popov criterion continuous, frequency-domain theorem


Lecture 25 - Popov criterion continuous, frequency-domain theorem


Lecture 26 - Describing function method


Lecture 27 - Describing Function : 2


Lecture 28 - Describing : optimal gain


Lecture 29 - Describing : optimal gain


Lecture 30 - Describing functions : Jump Hysteresis


Lecture 31 - Describing functions : sufficient conditions for existence of periodic orbits non existence of periodic orbits


Lecture 32 - Describing functions for nonlinearities


Lecture 33 - Ideal relay with Hysteresis and dead zone


Lecture 34 - Dynamical systems on manifolds-1


Lecture 35 - Dynamical systems on manifolds-2