Limit cycles bifurcating from a degenerate center

TL;DR

This paper investigates the maximum number of limit cycles bifurcating from a degenerate center in cubic polynomial systems, employing second-order averaging and computational verification, establishing an upper bound of three limit cycles.

Contribution

It provides the first comprehensive second-order analysis of limit cycles bifurcating from a degenerate center without Hamiltonian or rational integrals in cubic systems.

Findings

Maximum of three limit cycles bifurcate from the degenerate center.

First second-order study for such bifurcations in this class of systems.

Use of computational tools to verify complex algebraic calculations.

Abstract

We study the maximum number of limit cycles that can bifurcate from a degenerate center of a cubic homogeneous polynomial differential system. Using the averaging method of second order and perturbing inside the class of all cubic polynomial differential systems we prove that at most three limit cycles can bifurcate from the degenerate center. As far as we know this is the first time that a complete study up to second order in the small parameter of the perturbation is done for studying the limit cycles which bifurcate from the periodic orbits surrounding a degenerate center (a center whose linear part is identically zero) having neither a Hamiltonian first integral nor a rational one. This study needs many computations, which have been verified with the help of the algebraic manipulator Maple.