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The Period Function of Hamiltonian Systems with Separable Variables

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Abstract

In this paper we study the period function of those planar Hamiltonian differential systems for which the Hamiltonian function H(x, y) has separable variables, i.e., it can be written as \(H(x,y)=F_1(x)+F_2(y)\). More concretely we are concerned with the search of sufficient conditions implying the monotonicity of the period function, i.e., the absence of critical periodic orbits. We are also interested in the uniqueness problem and in this respect we seek conditions implying that there exists at most one critical periodic orbit. We obtain in a unified way several sufficient conditions that already appear in the literature, together with some other results that to the best of our knowledge are new. Finally we also investigate the limit of the period function as the periodic orbits tend to the boundary of the period annulus of the center.

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Acknowledgements

The first author is partially supported by MTM2017-86795-C3-2-P. The second author is partially supported by NNSF of China Grants 11671254 and 11871334. The first author would like to thank the School of Mathematical Sciences of the Shanghai Jiao Tong University in People’s Republic of China for the kind hospitality during his research stay.

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Authors and Affiliations

  1. Departament d’Enginyeria Informàtica i Matemàtiques, ETSE, Universitat Rovira i Virgili, 43007, Tarragona, Spain

    Jordi Villadelprat

  2. School of Mathematical Sciences, MOE–LSC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Xiang Zhang

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  1. Jordi Villadelprat
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  2. Xiang Zhang
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Correspondence to Xiang Zhang.

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Villadelprat, J., Zhang, X. The Period Function of Hamiltonian Systems with Separable Variables. J Dyn Diff Equat 32, 741–767 (2020). https://doi.org/10.1007/s10884-019-09759-w

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