Global Dynamics in Planar Piecewise Linear Hysteretic Systems

This paper investigates the global dynamics of planar piecewise linear hysteretic systems. First, we analyze the Hopf bifurcation in a general piecewise linear hysteretic systems, corresponding to the classification HLB17 in Simpsons taxonomy (Simpson, 2022, Physics Reports), and then deduced the stability criteria via the derivative of Poincaré map. Next, we examine a symmetric subclass of such systems, known as planar feedback systems, and prove that they admit at most one limit cycle. Furthermore, we provide a complete characterization of global phase portraits in the Poincaré disk, elucidating their topological structure. As an application, we construct a physical model consisting of a metal ball confined within a pipe and subjected to magnetic forces from symmetrically placed contacts. This setup yields a planar feedback system, and our analysis reveals two distinct regimes of periodic motion, depending on the relative strength of the magnetic interactions. A comprehensive bifurcation diagram is derived, delineating the transition between these dynamical behaviors.