CHAOTIC MOTION AND ITS PROBABILISTIC DESCRIPTION IN A FAMILY OF 2-DIMENSIONAL NONLINEAR-SYSTEMS WITH HYSTERESIS

Hysteresis-type nonlinearities often appear in engineering systems such as electric circuits, mechanical systems, and control systems. In this paper, we study a family of two-dimensional nonlinear dynamical systems that can be regarded as models for an active linear network or a linear control system with a hysteresis-type feedback. A complete analysis of the complicated chaotic behavior exhibited by such a system is presented in this paper. The Poincare return map is determined analytically, and a complete bifurcation study is completed in terms of two canonical parameters. The associated asymptotic behavior of the system is also discussed. Using tools from dynamical system and ergodic theory, a probabilistic description of the chaotic motion is obtained. We show that the chaotic system is isometric, from an ergodic point of view, to a time homogeneous Markov chain, for a certain set of canonical parameters.