scholarly journals Limit Cycle Bifurcations by Perturbing a Compound Loop with a Cusp and a Nilpotent Saddle

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Huanhuan Tian ◽  
Maoan Han
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Polynomial System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Melnikov Functions ◽  
Nilpotent Saddle

We study the expansions of the first order Melnikov functions for general near-Hamiltonian systems near a compound loop with a cusp and a nilpotent saddle. We also obtain formulas for the first coefficients appearing in the expansions and then establish a bifurcation theorem on the number of limit cycles. As an application example, we give a lower bound of the maximal number of limit cycles for a polynomial system of Liénard type.

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.

ASYMPTOTIC EXPANSIONS OF MELNIKOV FUNCTIONS AND LIMIT CYCLE BIFURCATIONS

2012 ◽  
Vol 22 (12) ◽  
pp. 1250296 ◽  
Author(s):  
MAOAN HAN
Keyword(s):  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Asymptotic Expansions ◽  
Melnikov Function ◽  
Heteroclinic Loop ◽  
First Order ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Center

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

2008 ◽  
Vol 18 (10) ◽  
pp. 3013-3027 ◽  
Author(s):  
MAOAN HAN ◽  
JIAO JIANG ◽  
HUAIPING ZHU
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Unperturbed System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Cubic System ◽  
Limit Cycle Bifurcation ◽  
Almost All ◽  
Nilpotent Center ◽  
Computing Method

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

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.

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.

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.

scholarly journals Limit Cycle Bifurcations Near a Cuspidal Loop

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1425
Author(s):  
Pan Liu ◽  
Maoan Han
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
First Order ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation

In this paper, we study limit cycle bifurcation near a cuspidal loop for a general near-Hamiltonian system by using expansions of the first order Melnikov functions. We give a method to compute more coefficients of the expansions to find more limit cycles near the cuspidal loop. As an application example, we considered a polynomial near-Hamiltonian system and found 12 limit cycles near the cuspidal loop and the center.

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