'American Institute of Mathematical Sciences (AIMS)'
Doi
Abstract
We consider an autonomous ordinary differential equation that admits a heteroclinic loop. The unperturbed heteroclinic loop consists of two degenerate heteroclinic orbits γ1 and γ2. We assume the variational equation along the degenerate heteroclinic orbit γi has {d_i}\left({{d_i} > 1, i = 1, 2} \right) linearly independent bounded solutions. Moreover, the splitting indices of the unperturbed heteroclinic orbits are s and −s(s≥0), respectively. In this paper, we study the persistence of the heteroclinic loop under periodic perturbation. Using the method of Lyapunov-Schmidt reduction and exponential dichotomies, we obtained the bifurcation function, which is defined from Rd1+d2+2 to Rd1+d2. Under some conditions, the perturbed system can have a heteroclinic loop near the unperturbed heteroclinic loop