Brake Orbits of a Reversible Even Hamiltonian System Near an Equilibrium

• 作者: Zhong Jie LIU Fan Jing WANG Duan Zhi ZHANG
• 来源期刊: 数学学报（英文版）
• 年，卷（期）: 2022,（1）
• 页码范围: 263-280
• 页数: 18页
• 语种: 英文

摘要

In this paper,we consider the brake orbits of a reversible even Hamiltonian system near an equilibrium.Let the Hamiltonian system(HS)(x)=JH′(x)satisfies H(0)=0,H′(0)=0,reversible and even conditions H(Nx)=H(x)and H(-x)=H(x)for all x ∈ R2n.Suppose the quadratic form Q(x)=1/2<H″(0)x,x>is non-degenerate.Fix τ0＞0 and assume that R2n=E ⊕ F decomposes into linear subspaces E and F which are invariant under the flow associated to the linear system x=JH″(0)x and such that each solution of the above linear system in E is r0-periodic whereas no solution in F \{0}is τ0-periodic.Write σ(τ0)=σQ(τ0)for the signature of Q|E.If σ(τ0)≠ 0,we prove that either there exists a sequence of brake orbits xk → 0 with τk-periodic on the hypersurface H-1(0)where τk → τ0;or for each λ close to 0 with λσ(τ0)＞0 the hypersurface H-1(λ)contains at least 1/2|σ(τ0)| distinct brake orbits of the Hamiltonian system(HS)near 0 with periods near τ0.Such result for periodic solutions was proved by Bartsch in 1997.