On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line

Abstract

In this paper, we investigate the existence and uniqueness of crossing limit cycle for a planar nonlinear Liénard system which is discontinuous along a straight line (called a discontinuity line). By using the Poincaré mapping method and some analysis techniques, a criterion for the existence, uniqueness and stability of a crossing limit cycle in the discontinuous differential system is established. An application to Schnakenberg model of an autocatalytic chemical reaction is given to illustrate the effectiveness of our result. We also consider a class of discontinuous piecewise linear differential systems and give a necessary condition of the existence of crossing limit cycle, which can be used to prove the non-existence of crossing limit cycle.

Crossing periodic orbits of nonsmooth Liénard systems and applications

“…Motivated by practical problems, many mathematicians and engineers have started to investigate the existence, uniqueness and number of limit cycles for the nonsmooth Liénard system (1.1). Usually, the nonsmoothness leads to much difficulty in the analysis of nonsmooth Liénard systems and much less results are obtained (see, e.g., [7,23,26,28]), compared with the smooth case. Consider Liénard system (1.1) with nonsmooth functions…”

“…Since the former one can be determined by one of the right and left systems, our attention is paid to the latter one, i.e., crossing limit cycle (see [24]). For (1.4), there exist a few results on the existence, uniqueness and number of crossing limit cycles, such as [7,23,26,28]. In [28], a necessary condition of the existence of crossing limit cycles and a sufficient condition for the uniqueness are given.…”

“…In [28], a necessary condition of the existence of crossing limit cycles and a sufficient condition for the uniqueness are given. In [23], a sufficient condition of the existence and uniqueness of crossing limit cycles is presented. The number of crossing limit cycles is studied in [7,26].…”

Stability Analysis of Equilibrium Point and Limit Cycle of Two-Dimensional Nonlinear Dynamical Systems—A Tutorial

“…A limit cycle is a closed trajectory in which the adjacent trajectories spiral toward or away from the limit cycle as time goes on [61]. Regarding the stability of the limit-cycle of a nonlinear dynamical system, we can use either LaSalle's local invariant set theorem to see if the limit cycle is stable or not [21], or we can use the Poincaré-Bendixson theorem to determine if there exist a stable limit cycle [62]. It is noted that the Poincaré-Bendixson theorem is only valid in two-dimensional systems.…”