# InfoCoBuild

## Nonlinear Dynamical Systems

Nonlinear Dynamical Systems. Instructors: Prof. Madhu N. Belur and Prof. Harish K. Pillai, Department of Electrical Engineering, IIT Bombay. This course covers basics of nonlinear differential equations that are encountered when dealing with practical dynamical systems in the context of their control. Topics covered include: introduction to nonlinear dynamical systems' features, existence and uniqueness of solutions, Lipschitz condition, linearization and local analysis, classification of equilibrium points (planar case), periodic orbits, Lyapunov theory, Lure problem, sector nonlinearity, Nyquist criterion, Lp stability, small gain theorem, passivity result, circle/Popov criteria, and describing function method. (from nptel.ac.in)

 Introduction

 Lecture 01 - Introduction Lecture 02 - First Order Systems Lecture 03 - Classification of Equilibrium Points Lecture 04 - Lipschitz Functions Lecture 05 - Existence and Uniqueness Theorems Lecture 06 - Existence and Uniqueness of Solutions to Differential Equations Lecture 07 - Lyapunov Theorem on Stability Lecture 08 - Extension of Lyapunov Theorem in Different Contexts Lecture 09 - LaSalle's Invariance Principle, Barbashin and Krasovski Theorems, Periodic Orbits Lecture 10 - Bendixson Criterion and Poincare-Bendixson Criterion 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 Lecture 13 - Signals and Norms of 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 and Nonlinearity, 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 Transformations and Circle Criterion Lecture 22 - Nonlinearities based on Circle Criterion Lecture 23 - Limit Cycles Lecture 24 - Popov Criterion, Frequency Domain Theorem Lecture 25 - Popov Criterion, Frequency Domain Theorem Lecture 26 - Describing Function Method Lecture 27 - Describing Function Lecture 28 - Describing Functions: Optimal Gain Lecture 29 - Describing Functions: Optimal Gain (cont.) Lecture 30 - Describing Functions: Jump Hysteresis Lecture 31 - Describing Functions: Sufficient Conditions for 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

 References Nonlinear Dynamical Systems Instructors: Prof. Madhu N. Belur and Prof. Harish K. Pillai, Department of Electrical Engineering, IIT Bombay. This course covers basics of nonlinear differential equations that are encountered when dealing with practical dynamical systems in the context of their control.