Markus–Yamabe Conjecture for Asymptotic Stability of Planar Piecewise Linear Systems

We present in this paper a detailed study on the Markus–Yamabe conjecture in planar piecewise linear systems. We consider discontinuous piecewise linear systems with two zones separated by a straight line, in which every subsystem is asymptotically stable. We prove the existence of limit cycles under explicit parameter conditions and give more different counterexamples to the Markus-Yamabe conjecture in addition to the counterexamples given by Llibre and Menezes. In particular, we consider continuous planar piecewise linear systems. For such a system with n + 1 zones separated by n parallel straight lines in phase space, we prove that if each of subsystems is asymptotically stable, then this system has a globally asymptotically stable equilibrium point, therefore the Markus–Yamabe conjecture still holds. Some examples are given to illustrate the main results.Mathematics Subject Classification (2020) 34C05 · 34C07 · 37G15

Limit Cycles and Bifurcations in a Class of Discontinuous Piecewise Linear Systems

In this paper, we consider planar piecewise linear differential systems with a line of discontinuity sharing a linear part. We study not only the number of crossing limit cycles, but also the number of sliding ones, and the coexistence of two configurations of limit cycles. In particular, we proved that both numbers of crossing limit cycles and sliding ones are at most [Formula: see text], but the total number of limit cycles is at most [Formula: see text]. Finally, by complete analysis on the number of limit cycles, we show some bifurcations which exist in generic Filippov systems, revealing also two nongeneric bifurcations.

On the Number of Invariant Cones and Existence of Periodic Orbits in 3-dim Discontinuous Piecewise Linear Systems

For a family of discontinuous 3-dim homogeneous piecewise linear dynamical systems with two zones, we investigate the number of invariant cones and the existence of periodic orbits as a spatial relationship between the invariant manifolds of the subsystem changes. By studying the number of real roots of a quadratic equation induced by slopes of half straight lines starting from the origin in required domain, we obtain complete results on the number and stability of invariant cones. Especially, we prove that the maximum number of invariant cones is two, and obtain complete parameter regions on which there exist one or two invariant cones, on which one or two fake cones (corresponding to real roots of the quadratic equation that are not in the required domain) appear and on which an invariant cone will be foliated by periodic orbits.