Limit Cycle Bifurcations in a Class of Piecewise Smooth Polynomial Systems

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Meilan Cai ◽  
Maoan Han
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Bifurcation Problem ◽  
First Order ◽  
Piecewise Polynomials ◽  
Straight Line ◽  
Cubic Systems

In this paper, we consider the bifurcation problem of limit cycles for a class of piecewise smooth cubic systems separated by the straight line [Formula: see text]. Using the first order Melnikov function, we prove that at least [Formula: see text] limit cycles can bifurcate from an isochronous cubic center at the origin under perturbations of piecewise polynomials of degree [Formula: see text]. Further, the maximum number of limit cycles bifurcating from the center of the unperturbed system is at least [Formula: see text] if the origin is the unique singular point under perturbations.

Bifurcation of a Kind of 1D Piecewise Differential Equation and Its Application to Piecewise Planar Polynomial Systems

2019 ◽  
Vol 29 (05) ◽  
pp. 1950072
Author(s):  
Jianfeng Huang ◽  
Yuye Jin
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Planar System ◽  
Order Analysis ◽  
First Order ◽  
Separation Curve ◽  
Straight Line ◽  
Melnikov Functions

This paper deals with a kind of piecewise smooth equation which is linear in the dependent variable. We study the problem of lower bounds for the maximum number of limit cycles of such equations using Melnikov functions. First of all, using the first order Melnikov function, we prove that these differential equations have a sharp upper bound for the number of the limit cycles which bifurcate from the periodic orbits and cross the separation straight line. Furthermore, in some cases the maximum number of these limit cycles is three, up to any order analysis. In the end, we apply this result on a kind of piecewise smooth planar system which has a separation curve with [Formula: see text] up to homomorphism.

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

2021 ◽  
Vol 31 (09) ◽  
pp. 2150123
Author(s):  
Xiaoyan Chen ◽  
Maoan Han
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Unperturbed System ◽  
Piecewise Polynomial ◽  
First Order ◽  
Number Of Zeros ◽  
Period Annulus

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].

Bifurcation Analysis in a Class of Piecewise Nonlinear Systems with a Nonsmooth Heteroclinic Loop

2018 ◽  
Vol 28 (02) ◽  
pp. 1850026
Author(s):  
Yuanyuan Liu ◽  
Feng Li ◽  
Pei Dang
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Heteroclinic Orbits ◽  
Phase Portraits ◽  
Unperturbed System ◽  
Piecewise Polynomial ◽  
Heteroclinic Loop ◽  
First Order ◽  
Nilpotent Critical Point

We consider the bifurcation in a class of piecewise polynomial systems with piecewise polynomial perturbations. The corresponding unperturbed system is supposed to possess an elementary or nilpotent critical point. First, we present 17 cases of possible phase portraits and conditions with at least one nonsmooth periodic orbit for the unperturbed system. Then we focus on the two specific cases with two heteroclinic orbits and investigate the number of limit cycles near the loop by means of the first-order Melnikov function, respectively. Finally, we take a quartic piecewise system with quintic piecewise polynomial perturbation as an example and obtain that there can exist ten limit cycles near the heteroclinic loop.

Limit Cycle Bifurcations Near a Piecewise Smooth Generalized Homoclinic Loop with a Saddle-Fold Point

2017 ◽  
Vol 27 (05) ◽  
pp. 1750071 ◽  
Author(s):  
Feng Liang ◽  
Dechang Wang
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Homoclinic Loop ◽  
First Order ◽  
Period Annulus ◽  
Straight Line ◽  
Smooth Perturbation

In this paper, we suppose that a planar piecewise Hamiltonian system, with a straight line of separation, has a piecewise generalized homoclinic loop passing through a Saddle-Fold point, and assume that there exists a family of piecewise smooth periodic orbits near the loop. By studying the asymptotic expansion of the first order Melnikov function corresponding to the period annulus, we obtain the formulas of the first six coefficients in the expansion, based on which, we provide a lower bound for the maximal number of limit cycles bifurcated from the period annulus. As applications, two concrete systems are considered. Especially, the first one reveals that a quadratic piecewise Hamiltonian system can have five limit cycles near a generalized homoclinic loop under a quadratic piecewise smooth perturbation. Compared with the smooth case [Horozov & Iliev, 1994; Han et al., 1999], three more limit cycles are found.

LIMIT CYCLE BIFURCATIONS OF TWO KINDS OF POLYNOMIAL DIFFERENTIAL SYSTEMS

2011 ◽  
Vol 21 (11) ◽  
pp. 3341-3357
Author(s):  
PEIPEI ZUO ◽  
MAOAN HAN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Function Method ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Bifurcation Problem ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Melnikov Function Method

In this paper, by using qualitative analysis and the first-order Melnikov function method, we consider two kinds of polynomial systems, and study the Hopf bifurcation problem, obtaining the maximum number of limit cycles.

Some Bifurcation Analysis in a Family of Nonsmooth Liénard Systems

2015 ◽  
Vol 25 (04) ◽  
pp. 1550055 ◽  
Author(s):  
Yuanyuan Liu ◽  
Maoan Han ◽  
Valery G. Romanovski
Keyword(s):  
Bifurcation Analysis ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Liénard Systems ◽  
Homoclinic Bifurcations

In this paper, we consider a class of piecewise smooth Liénard systems. We classify the unperturbed system into three types and study the bifurcation of limit cycles under perturbations. By studying the expansions of the first order Melnikov function, we give some new results on the number of limit cycles in homoclinic bifurcations.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

2020 ◽  
Vol 30 (15) ◽  
pp. 2050230
Author(s):  
Jiaxin Wang ◽  
Liqin Zhao
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Period Annulus ◽  
Quadratic Hamiltonian

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

Melnikov Method for a Class of Planar Hybrid Piecewise-Smooth Systems

2016 ◽  
Vol 26 (02) ◽  
pp. 1650030 ◽  
Author(s):  
Shuangbao Li ◽  
Wensai Ma ◽  
Wei Zhang ◽  
Yuxin Hao
Keyword(s):  
Melnikov Method ◽  
Homoclinic Solution ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Stable And Unstable Manifolds ◽  
Straight Line ◽  
Unstable Manifolds ◽  
Smooth System

In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed planar hybrid piecewise-smooth systems. In this class, the switching manifold is a straight line which divides the plane into two zones, and the dynamics in each zone is governed by a smooth system. When a trajectory reaches the separation line, then a reset map is applied instantaneously before entering the trajectory in the other zone. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise-smooth homoclinic solution transversally crossing the switching manifold. Then, we study the persistence of the homoclinic orbit under a nonautonomous periodic perturbation and the reset map. To achieve this objective, we obtain the Melnikov function to measure the distance of the perturbed stable and unstable manifolds and present the theorem for homoclinic bifurcations for the class of planar hybrid piecewise-smooth systems. Furthermore, we employ the obtained Melnikov function to detect the chaotic boundaries for a concrete planar hybrid piecewise-smooth system.

scholarly journals Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

2013 ◽  
Vol 2013 ◽  
pp. 1-19 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Melnikov Function ◽  
Phase Portraits ◽  
Unperturbed System ◽  
Linear Hamiltonian System ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Global Center

This paper concerns limit cycle bifurcations by perturbing a piecewise linear Hamiltonian system. We first obtain all phase portraits of the unperturbed system having at least one family of periodic orbits. By using the first-order Melnikov function of the piecewise near-Hamiltonian system, we investigate the maximal number of limit cycles that bifurcate from a global center up to first order ofε.

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