Persistence of the heteroclinic loop under periodic perturbation.

We consider an autonomous ordinary differential equation that admits a heteroclinic loop. The unperturbed heteroclinic loop consists of two degenerate heteroclinic orbits γ 1 and γ 2 . We assume the variational equation along the degenerate heteroclinic orbit γ i has d i ( d i > 1 , i = 1 , 2) 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 R d 1 + d 2 + 2 to R d 1 + d 2 . Under some conditions, the perturbed system can have a heteroclinic loop near the unperturbed heteroclinic loop. [ABSTRACT FROM AUTHOR]