Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems

TL;DR

This paper investigates minimal period problems for brake orbits in nonlinear autonomous reversible Hamiltonian systems, establishing lower bounds on periods and conditions for specific minimal periods based on the system's properties.

Contribution

It introduces new Maslov-type index theory applications to determine minimal periods of brake orbits in semipositive Hamiltonian systems, including even systems.

Findings

Existence of nonconstant T-periodic brake orbits with minimal period ≥ T/(2n+2).

Minimal period belongs to {T, T/2} if a certain matrix integral is positive definite.

Symmetric brake orbits in even systems have minimal periods in {T, T/3}.

Abstract

In this paper, for any positive integer $n$, we study the Maslov-type index theory of $i_{L_0}$, $i_{L_1}$ and $i_{\sqrt{-1}}^{L_0}$ with $L_0=\{0\}\times \R^n\subset \R^{2n}$ and $L_1=\R^n\times \{0\} \subset \R^{2n}$. As applications we study the minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltonian systems. For first order nonlinear autonomous reversible Hamiltonian systems in $\R^{2n}$, which are semipositive, and superquadratic at zero and infinity, we prove that for any $T>0$, the considered Hamiltonian systems possesses a nonconstant $T$ periodic brake orbit $X_T$ with minimal period no less than $\frac{T}{2n+2}$. Furthermore if $\int_0^T H"_{22}(x_T(t))dt$ is positive definite, then the minimal period of $x_T$ belongs to $\{T,\;\frac{T}{2}\}$. Moreover, if the Hamiltonian system is even, we prove that for any $T>0$, the considered even semipositive Hamiltonian systems possesses a nonconstant symmetric brake orbit with minimal period belonging to $\{T,\;\frac{T}{3}\}$