Limit Cycles for a Class of Piecewise Smooth Quadratic Differential Systems with Multiple Parameters

2016 ◽  
Vol 26 (10) ◽  
pp. 1650171 ◽  
Author(s):  
Xuekang Bo ◽  
Yun Tian
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Small Amplitude ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Isochronous Center ◽  
Differential Systems ◽  
First Order ◽  
Multiple Parameters ◽  
Perturbation Parameters

This paper considers a class of quadratic differential systems with an isochronous center under small piecewise smooth perturbations. Two perturbation parameters at different scales are included in the system. By using the first order Melnikov function, we obtain some new results on the number of small-amplitude limit cycles bifurcating around an isochronous center.

On the Limit Cycles Bifurcating from Piecewise Quasi-Homogeneous Differential Center

2016 ◽  
Vol 26 (07) ◽  
pp. 1650116 ◽  
Author(s):  
Shimin Li ◽  
Kuilin Wu
Keyword(s):  
Lower Bound ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Differential Systems ◽  
First Order

In this paper, a class of piecewise smooth quasi-homogeneous differential systems are considered. Using the first order Melnikov function derived in [Liu & Han, 2010], we obtain a lower bound of the maximum number of limit cycles which bifurcate from the periodic annulus of the center under polynomial perturbation. The results reveal that piecewise smooth quasi-homogeneous differential systems can bifurcate more limit cycles than the smooth systems.

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).

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.

scholarly journals On the Limit Cycles for Continuous and Discontinuous Cubic Differential Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Ziguo Jiang
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Quadratic Polynomial ◽  
Averaging Theory ◽  
Isochronous Center ◽  
Discontinuous Systems ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Cubic Polynomial

We study the number of limit cycles for the quadratic polynomial differential systemsx˙=-y+x2,y˙=x+xyhaving an isochronous center with continuous and discontinuous cubic polynomial perturbations. Using the averaging theory of first order, we obtain that 3 limit cycles bifurcate from the periodic orbits of the isochronous center with continuous perturbations and at least 7 limit cycles bifurcate from the periodic orbits of the isochronous center with discontinuous perturbations. Moreover, this work shows that the discontinuous systems have at least 4 more limit cycles surrounding the origin than the continuous ones.

HOPF BIFURCATION OF LIÉNARD SYSTEMS BY PERTURBING A NILPOTENT CENTER

2012 ◽  
Vol 22 (08) ◽  
pp. 1250203 ◽  
Author(s):  
JING SU ◽  
JUNMIN YANG ◽  
MAOAN HAN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Nonlinear Oscillators ◽  
Melnikov Function ◽  
Liénard System ◽  
First Order ◽  
Liénard Systems ◽  
Lienard System ◽  
Nilpotent Center

As we know, Liénard system is an important model of nonlinear oscillators, which has been widely studied. In this paper, we study the Hopf bifurcation of an analytic Liénard system by perturbing a nilpotent center. We develop an efficient method to compute the coefficients bl appearing in the expansion of the first order Melnikov function by finding a set of equivalent quantities B2l+1 which are able to calculate directly and can be used to study the number of small-amplitude limit cycles of the system. As an application, we investigate some polynomial Liénard systems, obtaining a lower bound of the maximal number of limit cycles near a nilpotent center.

scholarly journals Limit Cycles of a Class of Piecewise Smooth Liénard Systems

2016 ◽  
Vol 26 (01) ◽  
pp. 1650009 ◽  
Author(s):  
Lijuan Sheng
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Function Theory ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Piecewise Polynomial ◽  
Multiple Parameters ◽  
Liénard Systems ◽  
Limit Cycle Bifurcation

In this paper, we study the problem of limit cycle bifurcation in two piecewise polynomial systems of Liénard type with multiple parameters. Based on the developed Melnikov function theory, we obtain the maximum number of limit cycles of these two systems.

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.

Limit Cycles for a Class of Zp-Equivariant Differential Systems

2020 ◽  
Vol 30 (08) ◽  
pp. 2050115
Author(s):  
Jing Gao ◽  
Yulin Zhao
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Number Of Zeros ◽  
Planar Polynomial Differential Systems

In this paper, we study a class of [Formula: see text]-equivariant planar polynomial differential systems [Formula: see text]. It is shown that for any [Formula: see text] there is a differential system of the above type having at least [Formula: see text] limit cycles. This is proved by estimating the number of zeros of the first-order Melnikov function.

Limit Cycles from Hopf Bifurcation in Nongeneric Quadratic Reversible Systems with Piecewise Perturbations

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Chunyu Zhu ◽  
Yun Tian
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Melnikov Function ◽  
Reversible Systems ◽  
Piecewise Polynomial ◽  
Reversible System ◽  
First Order ◽  
Perturbed Systems

In this paper, we consider a nongeneric quadratic reversible system with piecewise polynomial perturbations. We use the expansion of the first order Melnikov function to obtain the maximal number of small-amplitude limit cycles produced by Hopf bifurcation in the perturbed systems.

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