Bifurcations and chaos in certain piecewise linear differential equations

Abstract

Piecewise linear differential equations of autonomous and nonautonomous types are ubiquitous systems modeling important nonlinear dynamical systems in physics and engineering. In particular, they have considerable relevance in the study of bifurcations and chaos in nonlinear electronics. Typical examples include the Chua's circuit, Murali-Lakshmanan-Chua circuit and the negative conductance forced series LCR circuit. In this article, we present a critical overview of some of these lower dimensional systems and show that they admit a wide variety of dynamical states including fixed points, limit cycles, bifurcations of different types to periodic orbits, quasiperiodic attractors, strange nonchaotic, chaotic and hyperchaotic attractors. The existence of these states is demonstrated using exact solutions and numerical analysis. These structures can also be demonstrated by experiments using electronic circuits. Controlling and synchronization in coupled arrays of such systems are also of great practical relevance. Finally we also discuss the bifurcation and chaos scenario in a typical piecewise linear scalar time delayed differential equation.