An approach toward higher dimensional hysteresis chaos generators

Abstract

Abstrnct -This paper discusses an approach toward higher dimensional autonomous chaotic circuits. We especially consider a particular class of circuits which includes only one nonlinear element, a three segments piecewise-linear resistor, and one small inductor La serially connected with it. The contents are divided into two parts. Part 1 gives a simple four- dimensional example that realizes hyperchaos. For the case where La is shorted, the circuit equation can be simplified into a constrained system and a two-dimensional Poincare map can be rigorously derived. This mapping and its Lyapunov exponents verify laboratory measurements of hyperchaos and related phenomena. Part 2 gives a rigorous approach to the singular perturbation theory of a N-dimensional circuit family which includes the example in Section I. We derive a canonical form equation which describes any circuit in this family. This equation can be simplified into a constrained system and a (N -2)-dimensional Poincare map can be derived. The main theorem indicates that this mapping explains the observ- able solutions of the canonical form very well.