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Homoclinic, heteroclinic and periodic orbits of singularly perturbed systems

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Abstract

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them comparing with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods. We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11671254) and Innovation Program of Shanghai Municipal Education Commission (Grant No. 15ZZ012). The author sincerely appreciates the referees for their nice comments and suggestions, which improved the presentation of this paper.

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  1. School of Mathematical Sciences, Key Laboratory of Scientific and Engineering Computing (Ministry of Education), Shanghai Jiao Tong University, Shanghai, 200240, China

    Xiang Zhang

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  1. Xiang Zhang
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Correspondence to Xiang Zhang.

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Zhang, X. Homoclinic, heteroclinic and periodic orbits of singularly perturbed systems. Sci. China Math. 62, 1687–1704 (2019). https://doi.org/10.1007/s11425-017-9223-6

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