On the Melnikov functions and limit cycles near a double homoclinic loop with a nilpotent saddle of order mˆ

2021 ◽  
Vol 291 ◽  
pp. 27-56
Author(s):  
Junmin Yang ◽  
Pei Yu ◽  
Maoan Han
Keyword(s):  
Limit Cycles ◽  
Homoclinic Loop ◽  
Melnikov Functions ◽  
Nilpotent Saddle

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 On Melnikov functions of a homoclinic loop through a nilpotent saddle for planar near-Hamiltonian systems

2008 ◽  
Vol 245 (4) ◽  
pp. 1086-1111 ◽  
Author(s):  
Hong Zang ◽  
Maoan Han ◽  
Dongmei Xiao
Keyword(s):  
Hamiltonian Systems ◽  
Homoclinic Loop ◽  
Melnikov Functions ◽  
Nilpotent Saddle

On the Number of Limit Cycles Bifurcated from a Near-Hamiltonian System with a Double Homoclinic Loop of Cuspidal Type Surrounded by a Heteroclinic Loop

2018 ◽  
Vol 28 (01) ◽  
pp. 1850004 ◽  
Author(s):  
Pegah Moghimi ◽  
Rasoul Asheghi ◽  
Rasool Kazemi
Keyword(s):  
Hamiltonian System ◽  
Lower Bound ◽  
Limit Cycles ◽  
Polynomial Systems ◽  
Homoclinic Loop ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
Nilpotent Saddle

In this paper, we study the number of bifurcated limit cycles from some polynomial systems with a double homoclinic loop passing through a nilpotent saddle surrounded by a heteroclinic loop, 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 in the following system: [Formula: see text] where [Formula: see text] is a polynomial of degree [Formula: see text].

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 Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Minzhi Wei ◽  
Junning Cai ◽  
Hongying Zhu
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Abelian Integral ◽  
Chebyshev System ◽  
Homoclinic Loop ◽  
Bifurcation Of Limit Cycles ◽  
Liénard System ◽  
Number Of Zeros ◽  
Lienard System ◽  
Nilpotent Saddle

In present paper, the number of zeros of the Abelian integral is studied, which is for some perturbed Hamiltonian system of degree 6. We prove the generating elements of the Abelian integral from a Chebyshev system of accuracy of 3; therefore there are at most 6 zeros of the Abelian integral.

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.

Bifurcation of Limit Cycles for Some Liénard Systems with a Nilpotent Singular Point

2015 ◽  
Vol 25 (05) ◽  
pp. 1550066 ◽  
Author(s):  
Junmin Yang ◽  
Xianbo Sun
Keyword(s):  
Singular Point ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Bifurcation Of Limit Cycles ◽  
Liénard Systems ◽  
Nilpotent Singular Point ◽  
Nilpotent Saddle

In this paper, we first present some general theorems on bifurcation of limit cycles in near-Hamiltonian systems with a nilpotent saddle or a nilpotent cusp. Then we apply the theorems to study the number of limit cycles for some polynomial Liénard systems with a nilpotent saddle or a nilpotent cusp, and obtain some new estimations on the number of limit cycles of these systems.

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.

On the Distribution of Limit Cycles in a Liénard System with a Nilpotent Center and a Nilpotent Saddle

2016 ◽  
Vol 26 (02) ◽  
pp. 1650025 ◽  
Author(s):  
R. Asheghi ◽  
A. Bakhshalizadeh
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Integral Formula ◽  
Abelian Integral ◽  
Liénard System ◽  
Algebraic Criterion ◽  
Period Annulus ◽  
Lienard System ◽  
Nilpotent Center ◽  
Nilpotent Saddle

In this work, we study the Abelian integral [Formula: see text] corresponding to the following Liénard system, [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are real bounded parameters. By using the expansion of [Formula: see text] and a new algebraic criterion developed in [Grau et al., 2011], it will be shown that the sharp upper bound of the maximal number of isolated zeros of [Formula: see text] is 4. Hence, the above system can have at most four limit cycles bifurcating from the corresponding period annulus. Moreover, the configuration (distribution) of the limit cycles is also determined. The results obtained are new for this kind of Liénard system.

scholarly journals Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Guoping Pang
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Sharp Bound ◽  
Abelian Integral ◽  
Heteroclinic Loop ◽  
Liénard System ◽  
Algebraic Criterion ◽  
Number Of Zeros ◽  
Lienard System ◽  
Nilpotent Saddle

In the presented paper, the Abelian integral I h of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.

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