Minimal period solutions in asymptotically linear Hamiltonian system with symmetries

Zhiping Fan; Duanzhi Zhang

doi:10.3934/dcds.2020354

Article Contents

2021, Volume 41, Issue 5: 2095-2124. Doi: 10.3934/dcds.2020354

This issue Previous Article Entropy conjugacy for Markov multi-maps of the interval Next Article Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems

Minimal period solutions in asymptotically linear Hamiltonian system with symmetries

Zhiping Fan^1, and
Duanzhi Zhang^2, ,

1.
School of Mathematical Sciences, Sun Yat-Sen University, Guangzhou 510300, China
2.
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

^* Corresponding author: Duanzhi Zhang
^* Corresponding author: Duanzhi Zhang

Received: October 31, 2019

Revised: April 30, 2020

Early access: October 2020

Published: May 2021

The first author is supported by the supported by the Fundamental Research Funds for the Central Universities (34000-31610273), Sun Yat-Sen University. The second author is supported by the NSF of China (17190271, 11422103, 11771341) and Nankai University

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Abstract

In this paper, applying the Maslov-type index theory for periodic orbits and brake orbits, we study the minimal period problems in asymptotically linear Hamiltonian systems with different symmetries. For the asymptotically linear semipositive even Hamiltonian systems, we prove that for any given $ T>0 $, there exists a central symmetric periodic solution with minimal period $ T $. Moreover, if the Hamiltonian systems are also reversible, we prove the existence of a central symmetric brake orbit with minimal period being either $ T $ or $ T/3 $. Also we give some other lower bound estimations for brake orbits case.

Keywords:

Minimal period,
Maslov-type indices,
Brake orbits,
Asymptotically linear Hamiltonian systems.

Mathematics Subject Classification: Primary: 37J12, 37J46; Secondary: 53D12.

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References

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