First-order Melnikov Function Research Articles

Limit Cycles from Perturbing a Piecewise Smooth System with a Center and a Homoclinic Loop

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.

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

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 Cycles for a Class of Zp-Equivariant Differential Systems

In this paper, we study a class of [Formula: see text]-equivariant planar polynomial differential systems [Formula: see text]. It is shown that for any [Formula: see text] there is a differential system of the above type having at least [Formula: see text] limit cycles. This is proved by estimating the number of zeros of the first-order Melnikov function.

A class of reversible quadratic systems with piecewise polynomial perturbations

This paper investigates a class of reversible quadratic systems perturbed inside piecewise polynomial differential systems of arbitrary degree n. All possible phase portraits of the reversible quadratic systems on the plane are obtained by analytic techniques. Then, we present the algebraic structure of its corresponding first-order Melnikov function, which can be used to study the limit cycle bifurcation problem. In this direction, we use it to investigate the Hopf bifurcation of the perturbed systems, and obtain an upper bound for the Hopf cyclicity. Meanwhile, we employ it to discuss the maximal number of limit cycles emerging from the periodic orbits around the origin for a special case and also provide its upper bound.

Bifurcation Analysis in a Class of Piecewise Nonlinear Systems with a Nonsmooth Heteroclinic Loop

We consider the bifurcation in a class of piecewise polynomial systems with piecewise polynomial perturbations. The corresponding unperturbed system is supposed to possess an elementary or nilpotent critical point. First, we present 17 cases of possible phase portraits and conditions with at least one nonsmooth periodic orbit for the unperturbed system. Then we focus on the two specific cases with two heteroclinic orbits and investigate the number of limit cycles near the loop by means of the first-order Melnikov function, respectively. Finally, we take a quartic piecewise system with quintic piecewise polynomial perturbation as an example and obtain that there can exist ten limit cycles near the heteroclinic loop.

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

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 cycles for perturbing Hamiltonian system inside piecewise smooth polynomial differential system

In this paper, we obtain the first-order Melnikov function of piecewise smooth polynomial perturbation of a Hamiltonian system. As application, we consider the number of limit cycles for perturbing the global center and truncated pendulum inside a piecewise smooth cubic polynomial differential system. Our results show that a piecewise smooth differential system can bifurcate more limit cycles than the smooth one.

Melnikov Method for a Three-Zonal Planar Hybrid Piecewise-Smooth System and Application

In this paper, we extend the well-known Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems, defined in three domains separated by two switching manifolds [Formula: see text] and [Formula: see text]. The dynamics in each domain is governed by a smooth system. When an orbit reaches the separation lines, then a reset map describing an impacting rule applies instantaneously before the orbit enters into another domain. We assume that the unperturbed system has a continuum of periodic orbits transversally crossing the separation lines. Then, we wish to study the persistence of the periodic orbits under an autonomous perturbation and the reset map. To achieve this objective, we first choose four appropriate switching sections and build a Poincaré map, after that, we present a displacement function and carry on the Taylor expansion of the displacement function to the first-order in the perturbation parameter [Formula: see text] near [Formula: see text]. We denote the first coefficient in the expansion as the first-order Melnikov function whose zeros provide us the persistence of periodic orbits under perturbation. Finally, we study periodic orbits of a concrete planar hybrid piecewise-smooth system by the obtained Melnikov function.

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

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 limit cycle bifurcation of a polynomial system from a global center

In this paper, we consider a quadratic system with a global center under polynomial perturbations of degree n(n ≥ 1). By using the first-order Melnikov function, we study Poincaré and Poincaré–Andronov–Hopf bifurcations. We prove that both Poincaré and Hopf cyclicity are n for n ≥ 2 up to the first order in ε.

Bifurcation of limit cycles from a Liénard system with a heteroclinic loop connecting two nilpotent saddles

In this article, we study the limit cycles bifurcated from a Lienard system with a heteroclinic loop connecting two nilpotent saddles. We apply expansion theory of a first-order Melnikov function to investigate the number of limit cycles near the heteroclinic loop and the center, and by some perturbation theory we find 3 limit cycles with 7 different distributions. Last, the least upper bound of the number of limit cycles bifurcated from the annulus is given by an algebraic criterion developed in J. Differ. Equ. 251, 1656–1669 (2011).

Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

This paper concerns limit cycle bifurcations by perturbing a piecewise linear Hamiltonian system. We first obtain all phase portraits of the unperturbed system having at least one family of periodic orbits. By using the first-order Melnikov function of the piecewise near-Hamiltonian system, we investigate the maximal number of limit cycles that bifurcate from a global center up to first order ofε.

Bifurcation of limit cycles in quadratic Hamiltonian systems with various degree polynomial perturbations

In this paper, we consider bifurcation of limit cycles in planar quadratic Hamiltonian systems with various degree polynomial perturbations. Attention is focused on the limit cycles which may appear in the vicinity of an isolated center, and up to 20th-degree polynomial perturbations are investigated. Restricted to the first-order Melnikov function, the method of focus value computation is used to determine the maximal number, H2(n), of small-amplitude limit cycles which may exist in the neighborhood of such a center. Besides the existing results H2(2)=2 and H2(3)=5, we shall show that H2(n)=43(n+1) for n=3,4,…,20.

BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A HYPER-ELLIPTIC HAMILTONIAN SYSTEM WITH TWO NILPOTENT SADDLES

In this paper we study the first-order Melnikov function for a planar near-Hamiltonian system near a heteroclinic loop connecting two nilpotent saddles. The asymptotic expansion of this Melnikov function and formulas for the first seven coefficients are given. Next, we consider the bifurcation of limit cycles in a class of hyper-elliptic Hamiltonian systems which has a heteroclinic loop connecting two nilpotent saddles. It is shown that this system can undergo a degenerate Hopf bifurcation and Poincare bifurcation, which emerges at most four limit cycles in the plane for sufficiently small positive e. The number of limit cycles which appear near the heteroclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. Further more we give all possible distribution of limit cycles bifurcated from the period annulus.

LIMIT CYCLE BIFURCATIONS OF TWO KINDS OF POLYNOMIAL DIFFERENTIAL SYSTEMS

In this paper, by using qualitative analysis and the first-order Melnikov function method, we consider two kinds of polynomial systems, and study the Hopf bifurcation problem, obtaining the maximum number of limit cycles.

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

The first-order Melnikov function of a homoclinic loop through a nilpotent saddle for general planar near-Hamiltonian systems is considered. The asymptotic expansion of this Melnikov function and formulas for its first coefficients are given. The number of limit cycles which appear near the homoclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. An example is presented as an application of the main results.

Perturbations of non-Hamiltonian reversible quadratic systems with cubic orbits

This paper is concerned with degree n polynomial perturbations of a class of planar non-Hamiltonian reversible quadratic integrable system whose almost all orbits are cubics. We give an estimate of the number of limit cycles for such a system. If the first-order Melnikov function (Abelian integral) M 1 ( h ) is not identically zero, then the perturbed system has at most 5 for n = 3 and 3 n - 7 for n ⩾ 4 limit cycles on the finite plane. If M 1 ( h ) is identically zero but the second Melnikov function is not, then an upper bound for the number of limit cycles on the finite plane is 11 for n = 3 and 6 n - 13 for n ⩾ 4 , respectively. In the case when the perturbation is quadratic ( n = 2 ), there exists a neighborhood U of the initial non-Hamiltonian polynomial system in the space of all quadratic vector fields such that any system in U has at most two limit cycles on the finite plane. The bound for n = 2 is exact.

On Hopf cyclicity of planar systems with multiple parameters

In this work, we discuss the maximal number of limit cycles which appear under perturbations in Hopf bifurcations by using degenerate first-order Melnikov function with multiple parameters.

On Hopf Cyclicity of Planar Systems

We investigate the maximal number of limit cycles which appear under perturbations in Hopf bifurcations by using the first-order Melnikov function with multiple parameters.

LOOKING FOR CHAOS.

This paper derives an alternative approach to the Melnikov method, which greatly reduces the amount of algebra involved in higher-order calculations. To illustrate this, a particular system is studied for which such a higher-order analysis is necessary, due to an identically vanishing first-order Melnikov function. The results of a second-order calculation imply the existence of transverse homoclinic orbits and, consequently, the existence of a horseshoe.