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

Duanzhi Zhang

doi:10.3934/dcds.2015.35.2227

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 $\{T,\;\frac{T}{3}\}$

Highlights

As applications we study the minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltonian systems
For the even first order Hamiltonian system, in [51], the author of this paper studied the minimal period problem of semipositive even Hamiltonian system and gave a positive answer to Rabinowitz’s conjecture in that case
Boundary conditions and the Morse index of the corresponding functional defined via the Galerkin approximation method on the finite dimensional truncated space at its corresponding critical points

Summary

Introduction and main results

In [32], Liu have considered the strictly convex reversible Hamiltonian systems case and proved the existence of nonconstant brake orbit of (1.1) with minimal period belonging to {T, T /2} for any given T > 0. For T > 0 such that iL√0−1(B0) + ν√L0−1(B0) = 0, under the same assumptions of Theorem 1.4, the system (1.1) possesses a nonconstant symmetric brake orbit with minimal period belonging to {T, T /3}. B0, under the same condition of Theorem 1.5, the system (1.1) possesses a nonconstant symmetric brake orbit with minimal period belonging to {T, T /3}.

Maslov-type index theory associated with Lagrangian subspaces

A Since Cfollow forms: