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

On the Limit Cycles for a Class of Continuous Piecewise Linear Differential Systems with Three Zones

International Journal of Bifurcation and Chaos, 2015

Lima and Llibre [2012] have studied a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map, they proved that this class admits always a unique limit cycle, which is hyperbolic. The class studied in [Lima & Llibre, 2012] belongs to a larger set of planar continuous piecewise linear vector fields with three zones that can be separated into four other classes. Here, we consider some of these classes and we prove that some of them always admit a unique limit cycle, which is hyperbolic. However we find a class that does not have limit cycles.

On the limit cycles of a class of piecewise linear differential systems in with two zones

Mathematics and Computers in Simulation, 2011

We study the bifurcation of limit cycles from the periodic orbits of a four-dimensional center in a class of piecewise linear differential systems with two zones. Our main result shows that three is an upper bound for the number of limit cycles that bifurcate from a center, up to first order expansion of the displacement function. Moreover, this upper bound is reached. The main technique used is the averaging method.

Two Limit Cycles in Liénard Piecewise Linear Differential Systems

Journal of Nonlinear Science, 2018

Some techniques for studying the existence of limit cycles for smooth differential systems are extended to continuous piecewise-linear differential systems. Rigorous new results are provided on the existence of two limit cycles surrounding the equilibrium point at the origin for systems with three zones separated by two parallel straight lines without symmetry.

Three Limit Cycles in Discontinuous Piecewise Linear Differential Systems with Two Zones

2015

In this paper we study a planar piecewise linear differential system formed by two regions separated by a straight line so that one system has a real unstable focus and the other a virtual stable focus which coincides with the real one. This system was introduced by S.-M. Huan and X.-S. Yang in [8] who numerically showed that it can exhibit 3 limit cycles surrounding the real focus. This is the first example that a non–smooth piecewise linear differential system with two zones can have 3 limit cycles surrounding a unique equilibrium. We provide a rigorous proof of this numerical result.

About Limit Cycles in Continuous Piecewise Linear Differential Systems

2017

In 2012, Lima and Llibre in [3] have studied a class of planar continuous piecewise linear vector fields with three zones. This class can be separated in four other classes and they proved, using the Poincaré map, that this particular class admits always a unique hyperbolic limit cycle. Here, we extended this study for other classes. We proved that some of them also admit always a unique hyperbolic limit cycle, moreover, we find a class that does not have limit cycles and prove the appearance of two limit cycles with one of these cycles appear by perturbations of a period annulus.

On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems

Applied Mathematics and Computation, 2015

In this paper we consider the linear differential center (ẋ,ẏ) = (−y, x) perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0. We provide sufficient conditions to ensure the existence of a limit cycle bifurcating from the infinity. The main tools used are the Bendixson transformation and the averaging theory.

Existence of piecewise linear differential systems with exactly n limit cycles for all

Nonlinear Analysis: Theory, Methods & Applications, 2003

In this paper, we prove that the piecewise linear di erential systeṁ x = −y − (x);ẏ = x with = 0 and an odd piecewise linear periodic function of period 4, has exactly n limit cycles in the strip |x| 6 2(n + 1). Consequently, there are piecewise linear di erential systems having inÿnitely many limit cycles in the real plane. We also provide examples of piecewise linear di erential systems having exactly n limit cycles for all n ∈ N. ?