Uniqueness and Non-uniqueness of Limit Cycles for Piecewise Linear Differential Systems with Three Zones and No Symmetry

Abstract

Abstract. Some techniques for proving the existence and uniqueness of limit cycles for smooth differential systems, are extended to continuous piecewiselinear differential systems. Then we obtain new results on the existence and uniqueness of limit cycles for systems with three linearity zones and without symmetry. We also reprove existing results of systems with two linearity zones giving shorter and clearer proofs.

Saddle-node of limit cycles in planar piecewise linear systems and applications

“…These class of new generation oscillators have potential to model the behavior of synapse connections in neurons. In [34], the authors propose the following modification for the nonlinear flux-charge characteristic of the memristor oscillator appearing in [14], namely,…”

“…In [35], the authors prove the existence of a codimension-1 manifold of saddle-node of limit cycles in a family of planar continuous PWL systems with three zones and symmetry coming from a heteroclinic connection. In [34], the authors study the number of limit cycles and prove the existence of two hyperbolic limit cycles with different stability surrounding one equilibrium point in a non-symmetric family of PWL systems. Even when this configuration is close to a saddle-node bifurcation as, for instance, we can observe in Figure 1 c) of [34], where two limit cycles are close one another, the saddle-node bifurcation is not reported in this paper.…”

“…In [34], the authors study the number of limit cycles and prove the existence of two hyperbolic limit cycles with different stability surrounding one equilibrium point in a non-symmetric family of PWL systems. Even when this configuration is close to a saddle-node bifurcation as, for instance, we can observe in Figure 1 c) of [34], where two limit cycles are close one another, the saddle-node bifurcation is not reported in this paper. In [40], boundary equilibrium bifurcations in planar continuous PWL systems with two and three zones are studied.…”

“…As a second application, we consider the memristor oscillator [14]. For this electronic circuit, we analyze the version of the model that was considered in [34]. Although in [34] they find cases where two limit cycles exist, they do not focus their attention on the existence of the saddle-node bifurcation of limit cycles.…”

“…Remark 3. Statements (d) and (e) of Theorems 5 and 6 in [34] provide conditions about the traces and determinants of the coefficient matrices of the system, including the inequality t L · t R < 0, such that there exists at most one limit cycle, which precludes the presence of a saddle-node bifurcation of limit cycles. Note that conditions in [34] are not satisfied under the hypotheses of Theorem 2.1, as it is assumed that the trace of the coefficients matrix in central zone t C = −m is small enough in absolute value.…”