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 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.

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 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.

Bifurcation of Limit Cycles in a Near-Hamiltonian System with a Cusp of Order Two and a Saddle

2016 ◽  
Vol 26 (11) ◽  
pp. 1650180 ◽  
Author(s):  
Ali Bakhshalizadeh ◽  
Hamid R. Z. Zangeneh ◽  
Rasool Kazemi
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
First Order ◽  
Period Annulus

In this paper, the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp of order two and a hyperbolic saddle for a planar near-Hamiltonian system is given. Next, we consider the limit cycle bifurcations of a hyper-elliptic Liénard system with this kind of heteroclinic loop and study the least upper bound of limit cycles bifurcated from the period annulus inside the heteroclinic loop, from the heteroclinic loop itself and the center. We find that at most three limit cycles can be bifurcated from the period annulus, also we present different distributions of bifurcated limit cycles.

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.

Poincaré Bifurcation of Some Nonlinear Oscillator of Generalized Liénard Type Using Symbolic Computation Method

2018 ◽  
Vol 28 (08) ◽  
pp. 1850096 ◽  
Author(s):  
Hongying Zhu ◽  
Bin Qin ◽  
Sumin Yang ◽  
Minzhi Wei
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Nonlinear Oscillator ◽  
Melnikov Function ◽  
Computation Method ◽  
Rigorous Proof ◽  
Heteroclinic Loop ◽  
Period Annulus ◽  
Domain Iii ◽  
Weak Damping

In this paper, we study the Poincaré bifurcation of a nonlinear oscillator of generalized Liénard type by using the Melnikov function. The oscillator has weak damping terms. When the damping terms vanish, the oscillator has a heteroclinic loop connecting a nilpotent cusp to a hyperbolic saddle. Our results reveal that: (i) the oscillator can have at most four limit cycles bifurcating from the corresponding period annulus. (ii) There are some parameters such that three limit cycles emerge in the original periodic orbit domain. (iii) Especially, we give a rigorous proof that [Formula: see text] limit cycle(s) can emerge near the original singular loop and [Formula: see text] limit cycle(s) can emerge near the original elementary center with [Formula: see text].

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 from an Order-3 Nilpotent Center of Cubic Hamiltonian Systems Perturbed by Cubic Polynomials

2020 ◽  
Vol 30 (09) ◽  
pp. 2050126
Author(s):  
Li Zhang ◽  
Chenchen Wang ◽  
Zhaoping Hu
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
First Order ◽  
Cubic Polynomials ◽  
Nilpotent Center

From [Han et al., 2009a] we know that the highest order of the nilpotent center of cubic Hamiltonian system is [Formula: see text]. In this paper, perturbing the Hamiltonian system which has a nilpotent center of order [Formula: see text] at the origin by cubic polynomials, we study the number of limit cycles of the corresponding cubic near-Hamiltonian systems near the origin. We prove that we can find seven and at most seven limit cycles near the origin by the first-order Melnikov function.

Limit Cycle Bifurcations by Perturbing a Hamiltonian System with a 3-Polycycle Having a Cusp of Order One or Two

2018 ◽  
Vol 28 (03) ◽  
pp. 1850038
Author(s):  
Marzieh Mousavi ◽  
Hamid R. Z. Zangeneh
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Perturbed Systems

In this paper, we study the asymptotic expansion of the first order Melnikov function near a 3-polycycle connecting a cusp (of order one or two) to two hyperbolic saddles for a near-Hamiltonian system in the plane. The formulas for the first coefficients of the expansion are given as well as the method of bifurcation of limit cycles. Then we use the results to study two Hamiltonian systems with this 3-polycycle and determine the number and distribution of limit cycles that can bifurcate from the perturbed systems. Moreover, a sharp upper bound for the number of limit cycles bifurcated from the whole periodic annulus is found when there is a cusp of order one.

scholarly journals LIMIT CYCLE BIFURCATIONS FROM CENTERS OF SYMMETRIC HAMILTONIAN SYSTEMS PERTURBED BY CUBIC POLYNOMIALS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350043 ◽  
Author(s):  
ZHAOPING HU ◽  
BIN GAO ◽  
VALERY G. ROMANOVSKI
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Standard Form ◽  
Melnikov Function ◽  
Singular Points ◽  
Computationally Efficient ◽  
Cubic Polynomials ◽  
Nilpotent Center

We study cubic near-Hamiltonian systems obtained by perturbing a symmetric cubic Hamiltonian system with two symmetric singular points. First, following [Han, 2012], we develop a method to study the analytical property of the Melnikov function near the origin for near-Hamiltonian system having the origin as its elementary center or nilpotent center. A computationally efficient algorithm based on the method is established to systematically compute the coefficients of the Melnikov function. Then, we consider the symmetric singular points and present the conditions for one of them to be an elementary center or a nilpotent center. Under the condition for the singular point to be a center, we obtain the standard form of the Hamiltonian system near the center. Moreover, perturbing the symmetric cubic Hamiltonian systems by cubic polynomials we study limit cycles bifurcating from the center. Finally, perturbing the symmetric Hamiltonian system by symmetric cubic polynomials, we consider the number of limit cycles near one of the symmetric centers of the symmetric near-Hamiltonian system, which is the same as that of another center.

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