Derivation of a continuous time dynamic planar system with two unstable foci from a three-dimensional chaotic piecewise linear system

In this paper, we introduce a class of continuous time dynamical planar systems that is capable of generating attractors in the plane by means of the use of hysteresis and at least two unstable foci. This class of systems shows stretching and folding behavior due to unstable equilibria and hysteresis. Hysteresis is used to overwhelm the constraints on the behavior of planar systems. This class of systems is derived from three-dimensional piecewise linear systems that have two manifolds, one stable and the other unstable, to generate heteroclinic chaos. Two numerical examples are given accordingly to the developed theory.