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

Abstract

In this paper, for any positive integer n, we study the Maslov-type index theory of i L0 , i L1As 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 T 2n+2 . Furthermore if T 0 H ′′ 22 (x T (t))dt is positive definite, then the minimal period of x T belongs to {T, 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, T 3 }.

The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems

“…For the minimal estimate of brake orbits, Liu [16] have considered the strictly convex reversible Hamiltonian systems case and proved the existence of nonconstant brake orbit with minimal period belonging to {T, T/2} for any given T > 0. Lately, Zhang studied the nonlinear autonomous reversible Hamiltonian systems case and proved the existence of nonconstant brake orbit with minimal period no less than T 2n+2 in [30]. For the brake subharmonic solutions of first order nonautonomous Hamiltonian systems, the author of this paper and Liu in [13] proved that when the positive integers j and k satisfy the certain conditions, there exists a jT -periodic nonconstant brake solution z j such that z j and z kj are distinct.…”

Multiplicity of brake orbits on compact convex symmetric reversible hypersurfaces in R 2n for n ⩾4

“…Hence we can joint P 1 M 2 P 2 to M 2 by symplectic path preserving the nullity ν L 0 and ν L 1 . By Lemma 2.2 of [23], (2.5) holds. Since P j N k = N k P j for j = 1, 2.…”

“…We first briefly review the index function (i ω , ν ω ) and (i L j , ν L j ) for j = 0, 1, more details can be found in [14] and [16]. Following Theorem 2.3 of [23] we study the differences…”

Seifert Conjecture in the Even Convex Case

“…In [16] it was proved that (L 0 , L 1 )-concavity is only depending on the end matrix γ(τ ) of γ, and in [35] it was proved that the (L 0 , L 1 )-concavity of a symplectic path γ is a half of the (ε, L 0 , L 1 )-signature of γ(τ ). i.e., we have the following result.…”

“…By results in [32,33,35], we have the following lemmas 3.3-3.5 which will be used frequently in Section 4.…”