Nonlinear Dynamics

Abstract

Contents: predator-prey systems, Lyapunov stability, Lyapunov function, Center Manifold Reduction, Hopf bifurcation, logistic map, Feigenbaum cascade, chaos, fundamental solution matrix, Poincaré map, Floquet multipliers, stability of periodic solutions, bifurcations of periodic solutions

Objective

This lecture is intended for graduate and PhD students from engineering sciences and physics who are interested in the behaviour of nonlinear dynamical systems. The course makes the student familiar with nonlinear phenomena such as limit cylces, quasiperiodicity, bifurcations and chaos. These nonlinear phenomena occur in for instance biological, economical, celestial and electrical systems but only mechanical multibody systems will be taken as examples. With the theory explained in the course one is able to understand flutter instability of wings, stick-slip vibrations, post-buckling behaviour of frames and nonlinear control techniques. Exercises and examples during the course include: hunting motion of railway vehicles, forced oscillation of a nonlinear mass-spring system, instability of the Watt stream governor and symmetric and asymmetric buckling. Engineering practice as well as the standard engineering curriculum often does not exceed a linear analysis of nonlinear systems. The course pays special attention to indicate the limitations of a linear analysis. The aim of the course is to give the student a basic knowledge and understanding of nonlinear system behaviour and to provide analysis tools to analyze nonlinear dynamical systems.

Content

1. Introduction: Notation; Literature 2. Dynamical Systems: Continuous-time systems; Discrete-time systems; Limit sets; Lyapunov stability 3. Bifurcations of Equilibria: Center Manifold; Center manifold reduction; Definition of Bifurcation; Normal forms 4. Bifurcations of Fixed Points of Discrete-time Systems; Linearization around a fixed point; One-dimensional linear discrete-time systems; Stability of fixed points of nonlinear discrete-time systems; Bifurcations of fixed points with a single eigenvalue +1; Flip bifurcation (single eigenvalue -1); Naimark-Sacker bifurcation (complex eigenvalue through unit circle); The logistic map; Horseshoes & intermittency 5. Stability and Bifurcations of Periodic Solutions; Periodicity properties; Fundamental Solution Matrix; Stability of periodic solutions; The Poincaré map; Bifurcations of periodic solutions; Harmonic Balance Method

Resources