Order Melnikov Function Research Articles

On the number of limit cycles for a perturbed cubic reversible Hamiltonian system.

This paper is concerned with the limit cycle problem of a cubic reversible Hamiltonian system under perturbation of polynomials of degree n with a switching line x=0. The upper and lower bounds of the number of limit cycles are obtained using the first order Melnikov function and its expansion. The method for calculating the Melnikov function relies upon some iterative formulas, which differs from other approaches.

Bounding the number of limit cycles for perturbed piecewise linear Hamiltonian system

In this paper, we investigate the number of limit cycles for piecewise linear Hamiltonian system with a homoclinic loop under perturbations of piecewise smooth polynomials. By studying the number of zeros of the first order Melnikov function, we obtain the exact number of limit cycles bifurcating from the periodic annulus, which improves the results given by Liu and Xiong [13], Yang and Zhang [20].

Proof of two conjectures for perturbed piecewise linear Hamiltonian systems

In this paper, we study the number of limit cycles bifurcating from the centers of piecewise linear Hamiltonian systems having either a homoclinic loop or a heteroclinic loop under the perturbations of piecewise smooth polynomials. By investigating the Chebyshev properties of generating functions of the first order Melnikov functions, we obtain the sharp bounds of the number of limit cycles bifurcating from the periodic annuluses, which confirm the conjectures proposed by Liang, Han and Romanovski (2012) and Liang and Han (2016).

Bifurcation theory of limit cycles by higher order Melnikov functions and applications

In this paper, we study Poincaré, Hopf and homoclinic bifurcations of limit cycles for planar near-Hamiltonian systems. Our main results establish Hopf and homoclinic bifurcation theories by higher order Melnikov functions, obtaining conditions on upper bounds and lower bounds of the maximum number of limit cycles. As an application, we concern a cubic near-Hamiltonian system, and study Hopf and homoclinic bifurcations in detail, finding more limit cycles than [26].

Melnikov analysis of a perturbed integrable non-Hamiltonian system

This paper provides a completed Melnikov analysis of any order for a class of perturbed polynomial differential systems, where the unperturbed system is a cubic center with a straight line of singular points. The emphasis on any order Melnikov function is crucial because of two major reasons: one is that higher order Melnikov functions are not in general expressible explicitly, the other one is the essential difficulty of computing elliptic integrals due to the complexity of many iterations. The exact upper bounds of the number of limit cycles bifurcated from the period annulus under different polynomial perturbations are obtained by analyzing the algebraic structures from explicit expressions of any order Melnikov functions.

Double homoclinic bifurcations by perturbing a class of cubic Z2-equivariant polynomial systems with nilpotent singular points

This paper focuses on the study of the limit cycle bifurcation of a general cubic Z2-equivariant Hamiltonian differential system with two nilpotent singular points. Initially, the necessary and sufficient conditions for the occurrence of two nilpotent singularities are provided in the unperturbed system, along with a complete description of all possible phase portraits on the plane. Subsequently, using a combination of the Melnikov function method and various analytical techniques, we derive the approximate expansion of the first order Melnikov function at the interval endpoint, as well as formula for its coefficients. These results can be utilized to analyze double homoclinic loop bifurcation of the perturbed system, finding the lower bound of limit cycles.

A Simple Calculation of the First Order Melnikov Function for a Non-smooth Hamilton System

A Simple Calculation of the First Order Melnikov Function for a Non-smooth Hamilton System

Bifurcation of Limit Cycles by Perturbing Piecewise Linear Hamiltonian Systems with Piecewise Polynomials

In this paper, we study a class of piecewise smooth near-Hamiltonian systems with piecewise polynomial perturbations. We first give the expression of the first order Melnikov function, and then estimate the number of limit cycles bifurcated from periodic annuluses by Melnikov function method. In addition, we discuss the number of limit cycles that can appear simultaneously near both sides of a generalized homoclinic or generalized double homoclinic loop.

Limit cycles near a homoclinic loop connecting a tangent saddle in a perturbed quadratic Hamiltonian system

In this paper, we study bifurcation of limit cycles from a homoclinic loop connecting a saddle of tangent type for a quadratic Hamiltonian system perturbed by nth degree polynomials, n=1,2,…,13. The main tool is the asymptotic expansion of the related Abelian integral near the homoclinic loop, and the maximal number of independent coefficients gives exact number of limit cycles. Our aim is to obtain more limit cycles by exploring more coefficients in the asymptotic expansion. However, it is usually very difficult to obtain the coefficients of the terms with degree greater than or equal to 2 in the asymptotic expansion. To overcome the difficulty, we derive two auxiliary systems and investigate the expansions for the related Abelian integral. The coefficients of lower degree terms in the new asymptotic expansions are equivalent to those of higher degree terms in the original asymptotic expansion. We obtain n−1−n−24 limit cycles near the non-regular homoclinic loop and n−24 limit cycles near the center, and it totally has at least n−1 limit cycles, when n∈{1,2,…,13}. The cyclicity of period annulus is also estimated by the first order Melnikov functions for n=3.

Limit cycle bifurcations of near-Hamiltonian systems with multiple switching curves and applications

<p style='text-indent:20px;'>In the present paper, we are devoted to the study of limit cycle bifurcations in piecewise smooth near-Hamiltonian systems with multiple switching curves, obtaining a formula of the first order Melnikov function in general case. As an application, we give lower bounds of the number of limit cycles for a piecewise smooth near-Hamiltonian system with a closed switching curve passing through the origin under piecewise polynomial perturbations.</p>

THE NUMBER OF LIMIT CYCLES FROM ELLIPTIC HAMILTONIAN VECTOR FIELDS BY HIGHER ORDER MELNIKOV FUNCTIONS

In this paper, the perturbed Hamiltonian system $dH=\epsilon F_4+\epsilon^2F_3+\epsilon^3F_2+\epsilon^4F_1$, with $F_i$ the vector valued homogeneous polynomials of degree $i$. The Hamiltonian function is $H=y^2/2+U(X),$ where $U$ is a univariate polynomial of degree four without symmetry. By computing higher order Melnikov functions, the upper bounds for the number of limit cycles that bifurcate from $dH=0$ are deserved.

Maximum number of limit cycles bifurcating from the period annulus of cubic polynomial systems

This paper is devoted to the limit cycle bifurcation problem for some cubic polynomial systems, whose unperturbed systems have a period annulus and two invariant lines. Using the first order Melnikov function and Chebyshev criterion, we obtain the maximum number of limit cycles bifurcating from the period annulus. It improves a known result given by Sui and Zhao [Internat. J. Bifur. Chaos Appl. Sci. Engrg. 28 (2018), 1850063].

On the Number of Limit Cycles Bifurcating from the Linear Center with an Algebraic Switching Curve

This paper studies the family of piecewise linear differential systems in the plane with two pieces separated by a switching curve $y=x^{m}$, where $m>1$ is an arbitrary positive. By analysing the first order Melnikov function, we give an upper bound and an lower bound of the maximum number of limit cycles which bifurcate from the period annulus around the origin under polynomial perturbations of degree $n$. The results shows that the degree of switching curves affect the number of limit cycles.

On the number of limit cycles of a class of Liénard–Rayleigh oscillators

This paper is devoted to the study of limit cycle bifurcations for a class of cubic near-Hamiltonian systems, which are in the form of Liénard–Rayleigh oscillators. By using geometric techniques to the first order Melnikov functions, we provide a complete study on the maximum number of limit cycles in Poincaré bifurcation from a period annulus.

Third Order Melnikov Functions of a Cubic Center under Cubic Perturbations

In this paper, cubic perturbations of the integral system (1+x)2dH where H=(x2+y2)/2 are considered. Some useful formulae are deduced that can be used to compute the first three Melnikov functions associated with the perturbed system. By employing the properties of the ETC system and the expressions of the Melnikov functions, the existence of exactly six limit cycles is given. Note that there are many cases for the existence of third-order Melnikov functions, and some existence conditions are very complicated—the corresponding Melnikov functions are not presented.

Further study on Horozov-Iliev’s method of estimating the number of limit cycles

In the study of the number of limit cycles of near-Hamiltonian systems, the first order Melnikov function plays an important role. This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the number of zeros of the function.

Limit Cycle Bifurcations Near a Heteroclinic Loop with Two Nilpotent Cusps of General Order

In this paper, we study the expansion of the first order Melnikov function near a heteroclinic loop with two nilpotent cusps of general order. More precisely, the order of the two cusps is [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] [Formula: see text]. For general [Formula: see text] and [Formula: see text], we give the expansion of the first order Melnikov function and the formulas for the first few coefficients. We further give a general theorem on the number of limit cycles bifurcated from the heteroclinic loop. These results extend the existing results for [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text]. As an application, these results are applied to study the number of limit cycles near a heteroclinic loop with two cusps of different order.

The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems

This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function H=12y2+12x2−23x3+a4x4(a≠0) under two types of polynomial perturbations of degree m, respectively. It is proved that the Hamiltonian system perturbed in Liénard systems can have at least [3m−14] small limit cycles near the center, where m≤101, and that the related near-Hamiltonian system with general polynomial perturbations can have at least m+[m+12]−2 small-amplitude limit cycles, where m≤16. Furthermore, in any of the cases, the bounds for limit cycles can be reached by studying the isolated zeros of the corresponding first order Melnikov functions and with the help of Maple programs. Here, [·] represents the integer function.

Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

In this paper, we study limit cycle bifurcations for planar piecewise smooth near-Hamiltonian systems with nth-order polynomial perturbation. The piecewise smooth linear differential systems with two centers formed in two ways, one is that a center-fold point at the origin, the other is a center-fold at the origin and another unique center point exists. We first explore the expression of the first order Melnikov function. Then by using the Melnikov function method, we give estimations of the number of limit cycles bifurcating from the period annulus. For the latter case, the simultaneous occurrence of limit cycles near both sides of the homoclinic loop is partially addressed.

The Exact Bound of the Number of Limit Cycles Bifurcating from the Global Nilpotent Center with a Nonlinear Switching Curve

In this paper, we study the bifurcations of a class of planar Hamiltonian systems having a global nilpotent center under the perturbations of polynomials of degree [Formula: see text] with a nonlinear switching curve. We obtain the exact bound of the number of limit cycles bifurcating from the period annulus if the first order Melnikov function is not identically zero. We also give some examples to illustrate our results.