On the number of limit cycles in general planar piecewise linear systems

Much progress has been made in planar piecewise smooth dynamical systems. However there remain many important problems to be solved even in planar piecewise linear systems. In this paper, we investigate the number of limit cycles of planar piecewise linear systems with two linear regions sharing the same equilibrium. By studying the implicit Poincaré map induced by the discontinuity boundary, some cases when there exist at most 2 limit cycles is completely investigated. Especially, based on these results we provide an example along with numerical simulations to illustrate the existence of 3 limit cycles thus have a negative answer to the conjecture by M. Han and W. Zhang [11](J. Differ.Equations 248 (2010) 2399-2416) that piecewise linear systems with only two regions have at most 2 limit cycles.