Narrow Results

In this work we continue the study of the centers which are limits of linear type centers. It is proved that if a degenerate center has an inverse integrating factor V(x, y) with V(0, 0) ≠ 0, then this degenerate center is also the limit of linear type centers. Moreover, we show that the degenerate centers with characteristic directions that are the limits of degenerate centers without characteristic directions are also detectable, from a theoretical point of view, with the Bautin 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.

In this paper we deal with a centrally symmetric quintic near-Hamiltonian system whose unperturbed system is a simple cubic Hamiltonian system having a nilpotent center. We prove that the system can have ten limit cycles by using a homoclinic bifurcation method based on stability-change.

As we know, Liénard system is an important model of nonlinear oscillators, which has been widely studied. In this paper, we study the Hopf bifurcation of an analytic Liénard system by perturbing a nilpotent center. We develop an efficient method to compute the coefficients b_l appearing in the expansion of the first order Melnikov function by finding a set of equivalent quantities B_2l+1 which are able to calculate directly and can be used to study the number of small-amplitude limit cycles of the system. As an application, we investigate some polynomial Liénard systems, obtaining a lower bound of the maximal number of limit cycles near a nilpotent center.

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 ${𝒳}$. As formal integrability is not enough to characterize such a center we use a more general object, namely, formal inverse integrating factors $V$ of ${𝒳}$. Although by the existence of $V$ 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 $n\in \mathbb{N}$ with $n\ge 2$ associated to ${𝒳}$ which is invariant under orbital equivalency of ${𝒳}$. Besides the leading terms in the $(1,n)$-quasihomogeneous expansions that $V$ can have, we also prove the following: (i) If $n$ is even and there exists $V$ then ${𝒳}$ has a center; (ii) if $n=2$, the existence of $V$ characterizes all the centers; (iii) if there is a $V$ with minimum “vanishing multiplicity” at the singularity then, generically, ${𝒳}$ has a center.

In this paper, we characterize the global nilpotent centers of polynomial differential systems of the linear form plus cubic homogeneous terms.

From [Han et al., 2009a] we know that the highest order of the nilpotent center of cubic Hamiltonian system is $3$. In this paper, perturbing the Hamiltonian system which has a nilpotent center of order $3$ 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.

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 $n\in N$ 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.