Persistence of the heteroclinic loop under periodic perturbation

Bin Long; Shanshan Xu; Bin Long; Shanshan Xu

doi:10.3934/era.2023054

Electronic Research Archive

2023, Volume 31, Issue 2: 1089-1105. doi: 10.3934/era.2023054

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Persistence of the heteroclinic loop under periodic perturbation

Bin Long ^,,
Shanshan Xu

School of Mathematics & Data Science, Shaanxi University of Science and Technology Xi'an 710021, P. R. China

Received: 24 September 2022 Revised: 04 December 2022 Accepted: 04 December 2022 Published: 14 December 2022

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.
- heteroclinic orbit,
- heteroclinic loop,
- bifurcation,
- Lyapunov-Schmidt reduction,
- exponential dichotomies
Citation: Bin Long, Shanshan Xu. Persistence of the heteroclinic loop under periodic perturbation[J]. Electronic Research Archive, 2023, 31(2): 1089-1105. doi: 10.3934/era.2023054

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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.

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