Global Representation of General Solution and Canonical forms for Piecewise Linear Oscillators. The Case of a Discontinuous Nonlinearity.

Abstract

In this paper, we describe a construction method of global representaion of a general solution and Poincare mapping (or phase return mapping) from local exact solutions (or half return maps) for piecewise linear oscillators with discontinuous nonlinearity by using a pseudo-feedback approach. It is shown that for some piecewise linear systems, a general solution can be global represented by the form of superposition of linear system response and pseudo-feedback responses. Based on these global representations, we propose canonical forms of general solutions, steady states and poincare mapping. According to this formulation, phase return mapping can be expressed in a global analytical form depending on return time parameters. Finally, numerical simulations are performed, and the validity of the presented approach can be confirmed.