Orbital Reversibility of Planar Vector Fields

In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility have been given. The procedure is useful in the center problem because any nondegenerate and nilpotent center is orbitally reversible. Moreover, using this algorithm is possible to find degenerate centers which are orbitally reversible.

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.

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

Formal Inverse Integrating Factor and the Nilpotent Center Problem

We are interested in deepening the knowledge of methods based on formal power series applied to the nilpotent center problem of planar local analytic monodromic vector fields [Formula: see text]. As formal integrability is not enough to characterize such a center we use a more general object, namely, formal inverse integrating factors [Formula: see text] of [Formula: see text]. Although by the existence of [Formula: see text] it is not possible to describe all nilpotent centers strata, we simplify, improve and also extend previous results on the relationship between these concepts. We use in the performed analysis the so-called Andreev number [Formula: see text] with [Formula: see text] associated to [Formula: see text] which is invariant under orbital equivalency of [Formula: see text]. Besides the leading terms in the [Formula: see text]-quasihomogeneous expansions that [Formula: see text] can have, we also prove the following: (i) If [Formula: see text] is even and there exists [Formula: see text] then [Formula: see text] has a center; (ii) if [Formula: see text], the existence of [Formula: see text] characterizes all the centers; (iii) if there is a [Formula: see text] with minimum “vanishing multiplicity” at the singularity then, generically, [Formula: see text] has a center.

LIMIT CYCLE BIFURCATIONS FROM CENTERS OF SYMMETRIC HAMILTONIAN SYSTEMS PERTURBED BY CUBIC POLYNOMIALS

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.

ASYMPTOTIC EXPANSIONS OF MELNIKOV FUNCTIONS AND LIMIT CYCLE BIFURCATIONS

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.