Minimal [formula omitted]-symmetric periodic solutions of general Hamiltonian systems and delay differential equations

Ben-Xing Zhou,Chungen Liu,Zhan Zhou,Xiaofei Zhang

doi:10.1016/j.jde.2022.04.039

Minimal [formula omitted]-symmetric periodic solutions of general Hamiltonian systems and delay differential equations

Ben-Xing Zhou, Chungen Liu + Show 2 more

https://doi.org/10.1016/j.jde.2022.04.039

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Journal: Journal of Differential Equations	Publication Date: May 10, 2022
Citations: 2

Affiliation: Guangzhou University, Foshan University, Shanxi Datong University

#General Hamiltonian Systems #Delay Differential Equations + Show 8 more

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Abstract

For a skew-symmetric non-degenerate 2n×2n matrix J∈L2n(R) and a J-symplectic matrix M with MTJ−1M=J−1, in this paper we redefine the Maslov-type (J,M)-index for J-symplectic path and develop the iteration theory of the Maslov-type (J,M)-index theory. As applications we study the minimal period problems for M-symmetric orbits of nonlinear autonomous supper-quadratic general Hamiltonian systems, where M is a J-symplectic matrix satisfying Mk=I2n. Also we study the minimal periodic problem for some super-linear delay systems as applications, and we give a positive answer to the Rabinowitz-type minimal period problem for a kind of super-linear delay equations with the forms studied by Kaplan and Yorke in [12].

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