Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis

TL;DR

This paper introduces a simplified higher-order analysis method for studying bifurcations of small limit cycles in cubic integrable systems, demonstrating the potential for up to eleven limit cycles under perturbations.

Contribution

It presents a new, simpler approach to higher-order bifurcation analysis, extending the understanding of limit cycle bifurcations in cubic systems beyond previous methods.

Findings

Up to eleven limit cycles can bifurcate in the studied system.

The pattern of limit cycles is analyzed up to 39rd-order perturbations.

No more than eleven limit cycles are observed in the analysis.

Abstract

In this paper, we present a method of higher-order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations. This method is equivalent to higher-order Melinikov function approach used for studying bifurcation of limit cycles around a center but simpler. Attention is focused on planar cubic polynomial systems and particularly it is shown that the system studied by H. Zoladek in the article (Eleven small limit cycles in a cubic vector field, Nonlinearity 8, 843--860, 1995) can indeed have eleven limit cycles under perturbations at least up to $7$th order. Moreover, the pattern of numbers of limit cycles produced near the center is discussed up to $39$th-order perturbations, and no more than eleven limit cycles are found.