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

2017/08/25 by Yun Tian, Pei Yu, Tian, Yun +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Dynamical Systems (math.DS) #FOS: Mathematics #Lipid metabolism and biosynthesis

paper · pdf · doi:10.48550/arxiv.1708.07864

openalex publication_date 2017/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 7th order. Moreover, the pattern of numbers of limit cycles produced near the center is discussed up to 39th-order perturbations, and no more than eleven limit cycles are found.

Related

• Bifurcation of ten small-amplitude limit cycles by perturbing a quadratic Hamiltonian system
• Limit cycles bifurcating from a degenerate center
• Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center
• Ten limit cycles near a cubic homoclinic loop with a nilpotent cusp
• On the number of limit cycles bifurcating from the linear center with an algebraic switching curve