Persistence of hyperbolic tori in Hamiltonian systems

Abstract

We generalize the well-known result of Graff and Zehnder on the persistence of hyperbolic invariant tori in Hamiltonian systems by considering non-Floquet, frequency varying normal forms and allowing the degeneracy of the unperturbed frequencies. The preservation of part or full frequency components associated to the degree of non-degeneracy is considered. As applications, we consider the persistence problem of hyperbolic tori on a submanifold of a nearly integrable Hamiltonian system and the persistence problem of a fixed invariant hyperbolic torus in a non-integrable Hamiltonian system. r

Persistence of lower dimensional tori of general types in Hamiltonian systems

“…In the case that M 22 is everywhere non-singular, our results hold for arbitrary smooth matrices M 11 , M 12 = M 21 and apply to non-degenerate lower invariant tori of all types. Similar to [40], the results also allow multiple eigenvalues of JM 22 (or M 22 ), for example, when k = 0, i can be equal to j in NR).…”

“…Consider (1.1) and let λ 1 (ω), · · · , λ 2m (ω) be eigenvalues of JM 22 (ω). We assume the weak form of Melnikov's second non-resonance condition (1.3), i.e.,…”

“…As a simple example, we let M (ω) be any smooth, non-singular, real symmetric, (n + 2m) × (n + 2m) matrix on O such that M 22 Then the non-resonance condition NR) holds automatically, which therefore assures the persistence of n-tori when the perturbation is sufficiently small. With respect to normal degeneracy, an essential point of the result in part 2) of Theorem 1 is that possible normal degeneracy can be compensated by the overall non-degeneracy of M (ω).…”

“…Then y is a constant and z must be a quasi-periodic function with basic frequencies chosen from components of ω. Since NR) implies the condition (2.4) (see Lemma 5.1 in Section 5), i.e, none of the eigenvalues of JM 22 can rationally depend on the basic frequencies, it is easy to see that z must be a constant solution of the last equation above, i.e., (2.5)…”

Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems

“…And the proof of Graff's result was later given by Zehnder [19], who used implicit function techniques. More recently, Li and Yi [11] generalized the results of Graff and Zehnder on the persistence of hyperbolic invariant tori in Hamiltonian systems by allowing the degeneracy of the unperturbed Hamiltonians and they obtain the preservation of part or full components of tangent frequencies. They adopted the Fourier series expansion for normal form N, which is a new technique.…”