General study on limit cycle bifurcation near a double homoclinic loop

Abstract

In this paper, we study the bifurcation problem of limit cycles near a general double homoclinic loop. We establish a general theory to obtain a lower bound of the maximal number of limit cycles near the double homoclinic loop. As an application, we prove that a near-Hamiltonian system of the form x˙=y(y2−1)+ε∑i=0naix2i+1,y˙=−x has at least [52n] limit cycles for 1≤n≤56. This number is maximal that we can find so far for the system.