Stability and Limit Cycle Bifurcation for Two Kinds of Generalized Double Homoclinic Loops in Planar Piecewise Smooth Systems

2014 ◽  
Vol 24 (12) ◽  
pp. 1450153
Author(s):  
Feng Liang ◽  
Maoan Han
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bound ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Homoclinic Loop ◽  
Limit Cycle Bifurcation

In this paper, we present two kinds of generalized double homoclinic loops in planar piecewise smooth systems. For their stability a criterion is provided. Under nondegenerate conditions, we prove that for each case there are at most five limit cycles which can be bifurcated from the generalized double homoclinic loop. Especially, we construct two concrete systems to show that the upper bound can be achieved in both cases.

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.

Bifurcation of Periodic Orbits Crossing Switching Manifolds Multiple Times in Planar Piecewise Smooth Systems

2019 ◽  
Vol 29 (12) ◽  
pp. 1950160
Author(s):  
Zhihui Fan ◽  
Zhengdong Du
Keyword(s):  
Limit Cycle ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Smooth Curves ◽  
Switching Curve ◽  
Smooth System

In this paper, we discuss the bifurcation of periodic orbits in planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. We assume that the unperturbed system has either a limit cycle or a periodic annulus such that the limit cycle or each periodic orbit in the periodic annulus crosses every switching curve transversally multiple times. When the unperturbed system has a limit cycle, we give the conditions for its stability and persistence. When the unperturbed system has a periodic annulus, we obtain the expression of the first order Melnikov function and establish sufficient conditions under which limit cycles can bifurcate from the annulus. As an example, we construct a concrete nonlinear planar piecewise smooth system with three zones with 11 limit cycles bifurcated from the periodic annulus.

Bifurcation of Limit Cycles in Small Perturbation of a Class of Liénard Systems

2014 ◽  
Vol 24 (01) ◽  
pp. 1450004 ◽  
Author(s):  
Xianbo Sun ◽  
Hongjian Xi ◽  
Hamid R. Z. Zangeneh ◽  
Rasool Kazemi
Keyword(s):  
Limit Cycle ◽  
Small Perturbation ◽  
Limit Cycles ◽  
Upper Bound ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
Algebraic Criterion ◽  
Liénard Systems ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Saddle

In this article, we study the limit cycle bifurcation of a Liénard system of type (5,4) with a heteroclinic loop passing through a hyperbolic saddle and a nilpotent saddle. We study the least upper bound of the number of limit cycles bifurcated from the periodic annulus inside the heteroclinic loop by a new algebraic criterion. We also prove at least three limit cycles will bifurcate and six kinds of different distributions of these limit cycles are given. The methods we use and the results we obtain are new.

Limit Cycle Bifurcations by Perturbing a Piecewise Hamiltonian System with a Double Homoclinic Loop

2016 ◽  
Vol 26 (06) ◽  
pp. 1650103 ◽  
Author(s):  
Yanqin Xiong
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Melnikov Function ◽  
Analytic Method ◽  
Piecewise Polynomial ◽  
Bifurcation Problem ◽  
Homoclinic Loop ◽  
Limit Cycle Bifurcation

This paper is concerned with the bifurcation problem of limit cycles by perturbing a piecewise Hamiltonian system with a double homoclinic loop. First, the derivative of the first Melnikov function is provided. Then, we use it, together with the analytic method, to derive the asymptotic expansion of the first Melnikov function near the loop. Meanwhile, we present the first coefficients in the expansion, which can be applied to study the limit cycle bifurcation near the loop. We give sufficient conditions for this system to have [Formula: see text] limit cycles in the neighborhood of the loop. As an application, a piecewise polynomial Liénard system is investigated, finding six limit cycles with the help of the obtained method.

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 Cycles from Perturbing a Piecewise Smooth System with a Center and a Homoclinic Loop

2021 ◽  
Vol 31 (10) ◽  
pp. 2150159
Author(s):  
Ai Ke ◽  
Maoan Han
Keyword(s):  
Lower Bound ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Homoclinic Loop ◽  
First Order ◽  
Piecewise Smooth System ◽  
Period Annulus ◽  
Smooth System

We study bifurcations of limit cycles arising after perturbations of a special piecewise smooth system, which has a center and a homoclinic loop. By using the Picard–Fuchs equation, we give an upper bound of the maximum number of limit cycles bifurcated from the period annulus between the center and the homoclinic loop. Furthermore, by applying the method of first-order Melnikov function we obtain a lower bound of the maximum number of limit cycles bifurcated from the center.

Limit Cycles Near a Piecewise Smooth Generalized Homoclinic Loop with a Nonelementary Singular Point

2015 ◽  
Vol 25 (13) ◽  
pp. 1550176 ◽  
Author(s):  
Feng Liang ◽  
Junmin Yang
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Lower Bound ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Homoclinic Loop ◽  
Period Annulus

In this paper, we deal with limit cycle bifurcations by perturbing a piecewise smooth Hamiltonian system with a generalized homoclinic loop passing through a nonelementary singular point. We first give an expansion of the first Melnikov function corresponding to a period annulus near the generalized homoclinic loop. Then, based on the first coefficients in the expansion we obtain a lower bound for the maximal number of limit cycles bifurcated from the period annulus. As applications, two concrete systems are considered.

LIMIT CYCLE BIFURCATIONS NEAR A DOUBLE HOMOCLINIC LOOP WITH A NILPOTENT SADDLE

2012 ◽  
Vol 22 (08) ◽  
pp. 1250189 ◽  
Author(s):  
MAOAN HAN ◽  
JUNMIN YANG ◽  
DONGMEI XIAO
Keyword(s):  
General Theory ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Type A ◽  
Polynomial Systems ◽  
Homoclinic Loop ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Saddle ◽  
Hyperbolic Saddle

Homoclinic bifurcation is a difficult and important topic of bifurcation theory. As we know, a general theory for a homoclinic loop passing through a hyperbolic saddle was established by [Roussarie, 1986]. Then the method of stability-changing to find limit cycles near a double homoclinic loop passing through a hyperbolic saddle was given in [Han & Chen, 2000], and further developed by [Han et al., 2003; Han & Zhu, 2007]. For a homoclinic loop passing through a nilpotent saddle there are essentially two different cases, which we distinguish by cuspidal type and smooth type, respectively. For the cuspidal type a general theory was recently established in [Zang et al., 2008]. In this paper, we consider limit cycle bifurcation near a double homoclinic loop passing through a nilpotent saddle by studying the analytical property of the first order Melnikov functions for general near-Hamiltonian systems and obtain the conditions for the perturbed system to have 8, 10 or 12 limit cycles in a neighborhood of the loop with seven different distributions. In particular, for the homoclinic loop of smooth type, a general theory is obtained as a consequence. We finally consider some polynomial systems and find a lower bound of the maximal number of limit cycles as an application of our main results.

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