Persistence of the heteroclinic loop under periodic perturbation

Abstract

We consider an autonomous ordinary differential equation that admits a heteroclinic loop. The unperturbed heteroclinic loop consists of two degenerate heteroclinic orbits $\gamma_1$ and $\gamma_2$. We assume the variational equation along the degenerate heteroclinic orbit $\gamma_i$ 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 $-s$ $(s\geq 0)$, 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 $\mathbb{R}^{d_1+d_2+2}$ to $\mathbb{R}^{d_1+d_2}$. Under some conditions, the perturbed system can have a heteroclinic loop near the unperturbed heteroclinic loop