@article{Long2023, 
author = {Bin Long and Shanshan Xu},
title = {Persistence of the heteroclinic loop under periodic perturbation},
year = {2023},
journal = {Electronic Research Archive},
volume = {31},
number = {2},
pages = {1089-1105},
keywords = {heteroclinic orbit, heteroclinic loop, bifurcation, Lyapunov-Schmidt reduction, exponential dichotomies},
url = {https://www.sciopen.com/article/10.3934/era.2023054},
doi = {10.3934/era.2023054},
abstract = {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.}
}