The planar discontinuous piecewise linear refracting systems have at most one limit cycle

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles.The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus–focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.     

Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters

This paper is concerned with the problem of limit cycle bifurcation for piecewise smooth near-Hamiltonian systems with multiple parameters. By the first Melnikov function, some novel criteria have been established for the existence of multiple limit cycles. Furthermore, an example is included to validate the obtained results by considering the maximum number of limit cycles for a piecewise quadratic system studied in Llibre and Mereu (2014) [12]. Compared with the result in the above reference, one more limit cycle is found by our method.     

Hopf bifurcation at infinity in 3D symmetric piecewise linear systems. Application to a Bonhoeffer–van der Pol oscillator

In this work, a Hopf bifurcation at infinity in three-dimensional symmetric continuous piecewise linear systems with three zones is analyzed. By adapting the so-called closing equations method, which constitutes a suitable technique to detect limit cycles bifurcation in piecewise linear systems, we give for the first time a complete characterization of the existence and stability of the limit cycle of large amplitude that bifurcates from the point at infinity. Analytical expressions for the period and amplitude of the bifurcating limit cycles are obtained. As an application of these results, we study the appearance of a large amplitude limit cycle in a Bonhoeffer–van der Pol oscillator.     

Melnikov Functions for a Class of Piecewise Hamiltonian Systems

This paper is concerned with the number of limit cycles for a class of piecewise Hamiltonian systems with two zones separated by two semi-straight lines. By constructing a Poincar\''{e} map, we obtain explicit expressions of the first, second and third order Melnikov functions. In addition, we apply their expressions to give upper bounds of the number of limit cycles bifurcated from a period annulus of a piecewise polynomial Hamiltonian system.     

On the existence of a stable limit cycle to a piecewise linear system in R2

The present paper is devoted to the existence of limit cycles of planar piecewise linear (PWL) systems with two zones separated by a straight line and singularity of type “focus-focus” and “focus-center.” Our investigation is a supplement to the classification of Freire et al concerning the existence and number of the limit cycles depending on certain parameters. To prove existence of a stable limit cycle in the case “focus-center,” we use a pure geometric approach. In the case “focus-focus,” we prove existence of a special configuration of five parameters leading to the existence of a unique stable limit cycle, whose period can be found by solving a transcendent equation. An estimate of this period is obtained. We apply this theory on a two-dimensional system describing the qualitative behavior of a two-dimensional excitable membrane model.     

Two Limit Cycles in Liénard Piecewise Linear Differential Systems

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. As a relevant application, it is shown the existence of bistable regimes in an asymmetric memristor-based electronic oscillator.

     

A Note on the Bendixson-Dulac Theorem for Refracted Systems with Multiple Zones

This note extends the Bendixson-Dulac theorem to refracted systems with multiple zones. As an application, we prove that piecewise linear Duffing-type system has neither crossing limit cycles nor sliding limit cycles Therefore, it gives a positive answer to the Conjecture of \cite{TA2007}.     

On the number of limit cycles by perturbing a piecewise smooth Hamilton system with two straight lines of separation

This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than in the case with one straight line of separation and more limit cycles can be bifurcated.     

The Maximum Number of Zeros of Functions with Parameters and Application to Differential Equations

In this paper, we first study the problem of finding the maximum number of zeros of functions with parameters and then apply the results obtained to smooth or piecewise smooth planar autonomous systems and scalar periodic equations to study the number of limit cycles or periodic solutions, improving some fundamental results both on the maximum number of limit cycles bifurcating from an elementary focus of order $k$ or a limit cycle of multiplicity $k$, or from a period annulus, and on the maximum number of periodic solutions for scalar periodic smooth or piecewise smooth equations as well.     

Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus. The sufficient and necessary conditions for existence and uniqueness of limit cycles are also given.     

一类异宿环分支极限环的唯一性

本文研究平面上一类两点或三点异宿环附近极限环的分支,在一简洁条件下证明了异宿环分支极限环的唯一性,并给出了极限环唯一存在的充要条件.作为对三维余维2分支的应用,解决了所出现的两点异宿环产生唯一极限环的问题.     

A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems

In the qualitative theory of differential equations in the plane one of the most difficult objects to study is the existence of limit cycles. There are many papers dedicated to this subject. Here we will present a survey mainly dedicated to the algebraic and explicit non-algebraic limit cycles of the polynomial differential systems in ${\mathbb{R}}^2$ and of the discontinuous piecewise differential systems in ${\mathbb{R}}^2$ formed by two linear differential systems separated by a straight line. For this class of discontinuous piecewise differential systems the study of their algebraic and explicit non-algebraic limit cycles just is starting. Here we provide the first explicit non-algebraic limit cycle for the discontinuous piecewise linear differential systems. Additionally we recall seven open questions related with these types of limit cycles.     

On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems.New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results.A relevant application to the McKean piecewise linear model of a single neuron activity is included.     

On the Number of Limit Cycles in Small Perturbations of a Piecewise Linear Hamiltonian System with a Heteroclinic Loop

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.     

一类平面多项式系统极限环的存在唯一性

研究一类平面2n 1次多项式微分系统的极限环问题,利用Hopf分枝理论得到了该系统极限环存在性与稳定性的若干充分条件,利用Cherkas和Zheilevych的唯一性定理得到了极限环唯一性的若干充分条件.     

Limit cycles of discontinuous piecewise differential systems formed by linear centers in R2 and separated by two circles

We show that discontinuous planar piecewise differential systems formed by linear centers and separated by two concentric circles can have at most three limit cycles. Usually is a difficult problem to provide the exact upper bound that a class of differential systems can exhibit. Here we also provide examples of such systems with zero, one, two, or three limit cycles.     

Limit cycles of piecewise linear dynamical systems with three zones and lateral systems

In this paper, we give some evidences what cause more limit cycles for piecewise dynamical systems. We say, the angles or the number of zones are critical points. We study an example of linear lateral systems and an example of linear Y-shape systems, and prove that they have five and four crossing limit cycles by using Newton-Kantorovich Theorem, respectively.     

Qualitative Analysis of Crossing Limit Cycles in Discontinuous Liénard-Type Differential Systems

In this paper, we investigate qualitative properties of crossing limit cycles for a class of discontinuous nonlinear Liénard-type differential systems with two zones separated by a straight line. Firstly, by applying left and right Poincaré mappings we provide two criteria on the existence, uniqueness and stability of a crossing limit cycle. Secondly, by geometric analysis we estimate the position of the unique limit cycle. Several lemmas are given to obtain an explicit upper bound for the amplitude of the limit cycle. Finally, a predatorprey model with nonmonotonic functional response is studied, and Matlab simulations are presented to show the agreement between theoretical results and numerical analysis.     

On Hopf bifurcation in non-smooth planar systems

As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).     

Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems

We discuss bifurcation of periodic orbits in discontinuous planar systems with discontinuities on finitely many straight lines intersecting at the origin and the unperturbed system has either a limit cycle or an annulus of periodic orbits. Assume that the unperturbed periodic orbits cross every switching line transversally exactly once. For the first case we give a condition for the persistence of the limit cycle. For the second case, we obtain the expression of the first order Melnikov function and establish sufficient conditions on the number of limit cycles bifurcate from the periodic annulus. Then we generalize our results to systems with discontinuities on finitely many smooth curves. As an application, we present a piecewise cubic system with 4 switching lines and show that the maximum number of limit cycles bifurcate from the periodic annulus can be affected by the position of the switching lines.