Limit cycle bifurcations of planar piecewise differential systems with three zones

This paper considers the limit cycle bifurcation problem of planar piecewise differential systems with three zones. Some computation formulas studied the problem of limit cycle bifurcations are provided by introducing multiple parameters. As an application to the obtained method, the number of limit cycles of a piecewise linear system with three zones studied in Lima et al. (2017) is discussed and some more limit cycles are found.     

一类三次系统的极限环

主要研究一类三次系统的极限环存在性问题,推广了C.Chicone[2]的结果,给出此类系统极限环存在定理.     

LIMIT CYCLES AND CENTER-FOCUS OF NONLINEAR SYSTEMS

The authors consider the nonhnear systems x=h(y)-F(x),y=-g(x) in which g(x) may be not differentiabte and the system can be nonsymmetric. Some conditions which en-sure that there exists an infinite number of limit cycles are obtained. A problem about center-focus put forward by R. Conti has been answered.     

Bifurcation of limit cycles in piecewise smooth systems via Melnikov function

In this short paper, we present some remarks on the role of the rstorder Melnikov functions in studying the number of limit cycles of piecewisesmooth near-Hamiltonian systems on the plane.     

Further studies on limit cycle bifurcations for piecewise smooth near-Hamiltonian systems with multiple parameters

This paper investigates the limit cycle bifurcations for piecewise smooth near-Hamiltonian systems with multiple parameters. The formulas for the second and third term in expansions of the first order Melnikov function are derived respectively. The main results improve some known conclusions.     

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.     

Bifurcation of limit cycles by perturbing piecewise non-Hamiltonian systems with nonlinear switching manifold

This paper is devoted to the study of limit cycles that can bifurcate of a perturbation of piecewise non-Hamiltonian systems with nonlinear switching manifold. We derive the first order Melnikov function to these systems. As application, the sharp upper bound of the number of bifurcated limit cycles of two concrete systems, whose switching manifolds are algebraic curves, is presented.     

Limit cycles for two classes of planar polynomial differential systems with uniform isochronous centers

In this article, we study the maximum number of limit cycles for two classes of planar polynomial differential systems with uniform isochronous centers. Using the first-order averaging method, we analyze how many limit cycles can bifurcate from the period solutions surrounding the centers of the considered systems when they are perturbed inside the class of homogeneous polynomial differential systems of the same degree. We show that the maximum number of limit cycles, $m$ and $m+1$, that can bifurcate from the period solutions surrounding the centers for the two classes of differential systems of degree $2m$ and degree $2m+1$, respectively. Both of the bounds can be reached for all $m$.     

平面折射系统极限环的存在和唯一性

本文考虑平面折射系统的极限环个数问题.根据左、右子系统的动力学性态,可以将其分为如下6种类型:焦点-焦点、焦点-鞍点、焦点-结点、鞍点-鞍点、鞍点-结点和结点-结点.利用Poincaré映射,本文证明折射系统为焦点-结点情形时最多存在1个极限环.     

Phase portraits of planar piecewise linear refracting systems: Focus-saddle case

This paper deals with planar piecewise linear refracting systems with a straight line of separation. Using the Poincaré compactification, we provide the classification of the phase portraits in the Poincaré disc of piecewise linear refracting systems with focus-saddle dynamics.     

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.     

一类平面多项式微分系统极限环的存在唯一性

本文研究一类平面多项式微分系统的极限环，得到了系统极限环存在、唯一的充分条件。     

Limit cycles for cubic systems with a symmetry of order 4 and without infinite critical points

In this paper we study those cubic systems which are invariant under a rotation of radians. They are written as where is complex, the time is real, and , are complex parameters. When they have some critical points at infinity, i.e. , it is well-known that they can have at most one (hyperbolic) limit cycle which surrounds the origin. On the other hand when they have no critical points at infinity, i.e. there are examples exhibiting at least two limit cycles surrounding nine critical points. In this paper we give two criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds the origin. Our results apply to systems having a limit cycle that surrounds either 1, 5 or 9 critical points, the origin being one of these points. The key point of our approach is the use of Abel equations.

     

Limit cycles of quadratic systems

In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert’s Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper, we present a proof of our earlier conjecture that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is (3:1) [V.A. Gaiko, Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Kluwer, Boston, 2003]. Besides, applying the Wintner–Perko termination principle for multiple limit cycles to our canonical system, we prove in a different way that a quadratic system has at most three limit cycles around a singular point (focus) and give another proof of the same conjecture.     

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.     

Invariant manifolds of periodic orbits for piecewise linear three-dimensional systems

This paper analyses the existence of invariant manifolds ofperiodic orbits for a specific piecewise linear three-dimensionalsystem with two zones, whose linear parts share a pair of imaginaryeigenvalues. This degenerate situation is obtained from thelack of controllability. The analysis proceeds by its reductionto a periodic one-dimensional equation for which some resultsof the Ambrosetti–Prodi type are given.     

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 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.