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 ${\gamma }_1$ 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 $-s$ $(s\ge 0)$, 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.