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- Title
Brake Orbits of a Reversible Even Hamiltonian System Near an Equilibrium.
- Authors
Liu, Zhong Jie; Wang, Fan Jing; Zhang, Duan Zhi
- 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 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 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) = 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 (τ 0) | 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.
- Subjects
HAMILTONIAN systems; EQUILIBRIUM; QUADRATIC forms; LINEAR systems; INVARIANT subspaces
- Publication
Acta Mathematica Sinica, 2022, Vol 38, Issue 1, p263
- ISSN
1439-8516
- Publication type
Article
- DOI
10.1007/s10114-022-0473-3