On the Number of Limit Cycles Bifurcated from a Near-Hamiltonian System with a Double Homoclinic Loop of Cuspidal Type Surrounded by a Heteroclinic Loop

In this paper, we study the number of bifurcated limit cycles from some polynomial systems with a double homoclinic loop passing through a nilpotent saddle surrounded by a heteroclinic loop, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems. In particular, we study the bifurcation of limit cycles in the following system: <(x) over dot> = y, <(y) over dot> = x(3) (x(2) - 1) (x(2) - 4) + epsilon f(x)y, where f(x) is a polynomial of degree 8 <= n <= 10.