Narrow Results

This paper is concerned with the bifurcation of limit cycles from a quintic system with one center. By using the averaging theory, we show that under any small quintic homogeneous perturbations, up to order 1 in ε, at most three limit cycles bifurcate from periodic orbits of the considered system, and this upper bound can be reached. Up to order 2 in ε, at most seven limit cycles emerge from periodic orbits of the unperturbed one.

This paper deals with the bifurcation of limit cycles for a quintic system with one center. Using the averaging method, we explain how limit cycles can bifurcate from the periodic annulus around the center of the considered system by adding perturbed terms which are the sum of homogeneous polynomials of degree $4k+1$ for $2\le k\le n$. We show that up to first-order averaging, at most five limit cycles can bifurcate from the period annulus of the unperturbed system for $n=2$, at most $5n-7$ limit cycles can bifurcate from the periodic annulus of the unperturbed system for any $n\ge 3$, and the upper bound is sharp for $n=2$ and for $n=3$.