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.

Keywords

• heteroclinic orbit
• heteroclinic loop
• bifurcation
• lyapunov-schmidt reduction
• exponential dichotomies