Brake Orbits of a Reversible Even Hamiltonian System Near an Equilibrium

Abstract

In this paper, we consider the brake orbits of a reversible even Hamiltonian system near an equilibrium. Let the Hamiltonian system (HS) ẋ = 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 $Q(x) = {1 \over 2}\left\langle {{H^{\prime \prime}}(0)x,x} \right\rangle$ 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 ẋ = JH″(0)x and such that each solution of the above linear system in E is τ₀-periodic whereas no solution in F {0} is τ₀-periodic. Write σ(τ₀) = σ_q(τ₀) for the signature of Q|_E. If σ(τ₀) ≠ 0, we prove that either there exists a sequence of brake orbits x_k → 0 with τ_k-periodic on the hypersurface H⁻¹(0) where τ_k → τ₀; or for each λ close to 0 with λσ(τ₀) > 0 the hypersurface H⁻¹(λ) contains at least ${1 \over 2}\left| {\sigma ({\tau _0})} \right|$ 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.