Limit Cycles Bifurcating from a Period Annulus in Continuous Piecewise Linear Differential Systems with Three Zones

Abstract

We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits at least two limit cycles that appear by perturbations of a period annulus. Moreover, we describe the bifurcation of the limit cycles for this class through two examples of two-parameter families of piecewise linear vector fields with three zones.

Two Limit Cycles in Liénard Piecewise Linear Differential Systems

“…The quoted authors are able to show the existence of two limit cycles surrounding the only equilibrium under adequate hypotheses. Similar results have been recently obtained in [19] by considering perturbations of systems without sign-symmetric traces but under rather non-generic hypotheses. In all the quoted cases, the location of the equilibrium is out of the central zone for having two limit cycles.…”

“…Since T C = 0 and v < u, we have a circular period annulus tangent to the line x = v and totally contained in the band −u < −v ≤ x ≤ v. The most external periodic orbit, which is tangent to the line x = v passes through the point (v, 0). Recalling that the dynamics in the right hand zone x > v is given by (19)…”

Limit cycles appearing from the perturbation of differential systems with multiple switching curves

“…Akhmet and Arugaslan [1] generalized the problem of Hopf bifurcation for a planar non-smooth system by considering discontinuities on finitely many nonlinear curves emanating from a vertex. For more results, one can see [6,14,16,20,23,24] and the references therein.…”

Limit Cycles Bifurcating from a Periodic Annulus in Discontinuous Planar Piecewise Linear Hamiltonian differential System with Three Zones

“…More precisely, for symmetric continuous piecewise linear differential systems with three zones, conditions for nonexistence and existence of one, two or three limit cycles have been obtained (see for instance the book [15]). For the nonsymmetric case, examples with two limit cycles surrounding the only singular point at the origin was found in [13,16].…”