Minimal $ P $-symmetric periodic solutions in semi-positive definite Hamiltonian systems

Peng Liu; Duanzhi Zhang; Zhihao Zhao

doi:10.3934/dcds.2024029

Article Contents

2024, Volume 44, Issue 8: 2327-2341. Doi: 10.3934/dcds.2024029

This issue Previous Article Optimal control of a stationary Navier-Stokes hemivariational inequality with numerical approximation Next Article Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium type equation

Minimal $ P $-symmetric periodic solutions in semi-positive definite Hamiltonian systems

1.
School of Mathematical Sciences, Nankai University, Tianjin 300071, China
2.
Great Bay Institute for Advanced Study, Dongguan 523000, China
3.
University of Science and Technology of China, Hefei 230026, China
4.
School of Sciences, Great Bay University, Dongguan 523000, China

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

Received: November 2023

Revised: March 2024

Early access: March 2024

Published: August 2024

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Abstract

Let $ P $ be a real symplectic matrix satisfying $ P^k = I_{2n} $, where $ k: = \min\{m\in \mathbf{N}^{\ast}\mid P^m = I_{2n}\} $. In this paper, we study the minimal $ P $-symmetric period problem of autonomous and semi-positive definite Hamiltonian systems in $ \mathbf{R}^{2n} $. Applying the iterative inequality and Hörmander index estimate of $ P $-index, we prove that for every $ \tau>0 $, there exists a non-constant $ P $-solution with minimal $ P $-symmetric period not less than $ \frac{\tau} {2n+1} $.

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Mathematics Subject Classification: Primary: 70H05, 34C25; Secondary: 58E05, 53D12, 58J30.

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