Limit Cycle Bifurcations in Near-Hamiltonian Systems by Perturbing a Nilpotent Center

Abstract

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.

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

“…In the second case, the origin is called a nilpotent center. Limit cycle bifurcations near a nilpotent center and a cuspidal loop for general plane system were studied in [7] and [10] respectively. For the perturbation of a cuspidal loop of non-Hamiltonian systems, see [5] and the references therein.…”

On the number of limit cycles near a homoclinic loop with a nilpotent singular point

“…There are many works discussing bifurcations relating to nilpotent singular point, see [6,14,15,17,24,31,50]. Assume that (1.2) has a nilpotent singular point at the origin, that is to say, the function H satisfies H x (0, 0) =H y (0, 0) = 0, and…”

Limit Cycle Bifurcations From Centers of Symmetric Hamiltonian Systems Perturbed by Cubic Polynomials

“…The algorithm was implemented in the computer algebra system Maple. Earlier, Han, Jiang & Zhu [2008] proved Theorem 2 for p > 1. Then following Han, Yang & Yu [2009], Han [2012] gave a new proof.…”

“…Obviously, for small ε system (1) has a limit cycle near the origin if and only if the function F (h, ε, δ) has an isolated positive zero in h near h = 0. Based on the analytical property of the Melnikove function M(h, δ) and the number of limit cycles near the origin by the function, we have the following theorems (see Han, Jiang & Zhu [2008]). …”