scholarly journals The number of limit cycles for a class of quintic Hamiltonian systems under quintic perturbations

2002 ◽  
Vol 73 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Guowei Chen ◽  
Yongbin Wu ◽  
Xinan Yang
Keyword(s):  
Hopf Bifurcation ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Homoclinic Bifurcation

AbstractThe Hopf bifurcation and homoclinic bifurcation of the quintic Hamiltonian system is analyzed under quintic perturbations by using unfolding theory in this paper. We show that a quintic system can have at least 29 limit cycles.

LIMIT CYCLES FOR A CLASS OF QUINTIC NEAR-HAMILTONIAN SYSTEMS NEAR A NILPOTENT CENTER

2009 ◽  
Vol 19 (06) ◽  
pp. 2107-2113 ◽  
Author(s):  
JIAO JIANG ◽  
JIZHOU ZHANG ◽  
MAOAN HAN
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Homoclinic Bifurcation ◽  
Unperturbed System ◽  
Bifurcation Method ◽  
Simple Cubic ◽  
Stability Change ◽  
Centrally Symmetric ◽  
Nilpotent Center

In this paper we deal with a centrally symmetric quintic near-Hamiltonian system whose unperturbed system is a simple cubic Hamiltonian system having a nilpotent center. We prove that the system can have ten limit cycles by using a homoclinic bifurcation method based on stability-change.

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.

SMALL LIMIT CYCLES BIFURCATING FROM Z4-EQUIVARIANT NEAR-HAMILTONIAN SYSTEM OF DEGREES 9 AND 7

2012 ◽  
Vol 22 (11) ◽  
pp. 1250272 ◽  
Author(s):  
XIANBO SUN ◽  
JUNMIN YANG
Keyword(s):  
Hopf Bifurcation ◽  
Hamiltonian System ◽  
Limit Cycles ◽  
Bifurcation Theory

In this paper, we study the number and distribution of small limit cycles of some Z4-equivariant near-Hamiltonian system of degree 9. Using the methods of Hopf bifurcation theory, we find that this system can have 64 small limit cycles. The configuration of 64 small limit cycles of the system is also illustrated in Fig. 1. When we let some parameters be zero, then we find that there can be 40 small limit cycles in a seventh system.

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.

Bifurcation of Multiple Limit Cycles in an Epidemic Model on Adaptive Networks

2019 ◽  
Vol 29 (07) ◽  
pp. 1950096 ◽  
Author(s):  
Xiaoguang Zhang ◽  
Zhen Jin ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Critical Point ◽  
Limit Cycles ◽  
Epidemic Model ◽  
Homoclinic Bifurcation ◽  
Adaptive Networks ◽  
Numerical Example

Very recently, Zhang et al. considered an epidemic model on adaptive networks [Zhang et al., 2019], in which Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation are studied. Degenerate Hopf bifurcation is investigated via simulation and a numerical example is given to show the existence of two limit cycles. However, whether the codimension of the Hopf bifurcation is two is still open. In this paper, we will rigorously prove that the codimension of the Hopf bifurcation is two. That is, the maximal two limit cycles can bifurcate from the Hopf critical point. Moreover, the conditions for the existence of two limit cycles are derived.

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.

Limit Cycles of a Perturbation of a Polynomial Hamiltonian Systems of Degree 4 Symmetric with Respect to the Origin

2019 ◽  
Vol 63 (3) ◽  
pp. 547-561
Author(s):  
Jaume Llibre ◽  
Paulina Martínez ◽  
Claudio Vidal
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Polynomial Systems ◽  
Averaging Theory ◽  
Quartic Polynomial

AbstractWe study the number of limit cycles bifurcating from the origin of a Hamiltonian system of degree 4. We prove, using the averaging theory of order 7, that there are quartic polynomial systems close these Hamiltonian systems having 3 limit cycles.

Limit Cycle Bifurcations by Perturbing a Hamiltonian System with a Cuspidal Loop of Order m

2015 ◽  
Vol 25 (06) ◽  
pp. 1550083 ◽  
Author(s):  
Yanqing Xiong
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Liénard System ◽  
First Order ◽  
Lienard System

This paper is concerned with the expansion of the first-order Melnikov function for general Hamiltonian systems with a cuspidal loop having order m. Some criteria and formulas are derived, which can be used to obtain first-order coefficients in the expansion. In particular, we deduce the first-order coefficients for the case m = 3 and give the corresponding conditions of existing several limit cycles. As an application, we study a Liénard system of type (n, 9) and prove that it can have 14 limit cycles near a cuspidal loop of order 3 for n = 8.

On the Number of Limit Cycles Bifurcated from Some Hamiltonian Systems with a Double Homoclinic Loop and a Heteroclinic Loop

2017 ◽  
Vol 27 (04) ◽  
pp. 1750055 ◽  
Author(s):  
Pegah Moghimi ◽  
Rasoul Asheghi ◽  
Rasool Kazemi
Keyword(s):  
Hamiltonian System ◽  
Lower Bound ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Homoclinic Loop ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
Nilpotent Saddle ◽  
Hyperbolic Saddle

In this paper, we study the number of bifurcated limit cycles from near-Hamiltonian systems where the corresponding Hamiltonian system has a double homoclinic loop passing through a hyperbolic saddle surrounded by a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems. In particular, we study the bifurcation of limit cycles of the following system [Formula: see text] as an application of our results, where [Formula: see text] is a polynomial of degree five.

scholarly journals Second Order Melnikov Functions of Piecewise Hamiltonian Systems

2020 ◽  
Vol 30 (01) ◽  
pp. 2050016
Author(s):  
Peixing Yang ◽  
Jean-Pierre Françoise ◽  
Jiang Yu
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Second Order ◽  
Quadratic Polynomial ◽  
First Order ◽  
Melnikov Functions

In this paper, we consider the general perturbations of piecewise Hamiltonian systems. A formula for the second order Melnikov functions is derived when the first order Melnikov functions vanish. As an application, we can improve an upper bound of the number of bifurcated limit cycles of a piecewise Hamiltonian system with quadratic polynomial perturbations.

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