Nonlinear Dynamics and Chaos

Abstract

This chapter presents an overview of nonlinear dynamics and chaos. It starts with a background revision of dynamical systems. Concepts of equilibrium points, linearization, stability, and Poincare maps are treated. Afterward, chaotic dynamics is explored. Horseshoe transformation is discussed in order to define the main aspects of chaos. Fractal characteristics are presented and discussed. Routes to chaos are investigated showing some definitions of bifurcation. Lyapunov exponents are defined presenting a diagnostic tool for chaos. The main concepts and tools are then presented by considering a case study related to a shape memory alloy system. Single and two degrees of freedom systems are treated using a polynomial constitutive model to describe the restitution forces.