Limit Cycles Appearing from a Generalized Heteroclinic Loop with an Elementary Saddle and a Nilpotent Saddle by Perturbing Piecewise Hamiltonian Systems

TL;DR

This paper analyzes how limit cycles bifurcate from a heteroclinic loop involving elementary and nilpotent saddles in near-Hamiltonian systems, deriving conditions for their emergence using Melnikov functions and Abelian integrals.

Contribution

It provides a detailed analysis of limit cycle bifurcations near heteroclinic loops with elementary and nilpotent saddles, including explicit bounds on the number of limit cycles.

Findings

At least 4[(n+1)/2]+1 limit cycles can exist under perturbations.

Phase portraits and conditions for heteroclinic loops are characterized.

Expansion of the Melnikov function near the heteroclinic loop is derived.

Abstract

In this paper, we study limit cycle bifurcations for a class of general near-Hamiltonian systems near a heteroclinic loop with an elementary saddle and a nilpotent saddle. Firstly, we consider the behaviors of the unperturbed system, providing the phase portraits of the system and the necessary conditions for the appearance of a heteroclinic loop with an elementary saddle and a nilpotent saddle by using the relevant qualitative theory. Then, with consideration of the expression of the first-order Melnikov function, we derive its expansion near the heteroclinic loop by employing some techniques and properties of the Abelian integral. Finally, we investigate the coefficients of the expansion and show that there can exist at least $4[\frac{n+1}{2}]+1$ limit cycles under disturbance.