On Invariant Tori of Nearly Integrable Hamiltonian Systems with Quasiperiodic Perturbation

Abstract

We are concerned with the persistence of frequency of invariant tori for analytic integrable Hamiltonian system with quasiperiodic perturbation. It is proved that if the unperturbed system satisfies the Rüssmann's nondegeneracy condition and has nonzero Brouwer's topological degree at some Diophantine frequency; the perturbed system satisfies the colinked nonresonant condition, then the invariant torus with this frequency persists under quasiperiodic perturbation.

The generalized von Neumann-Jordan type constant and fixed points for multivalued nonexpansive mappings

“…It is shown in that if C Z (X ) < 1 2 1 + 1 R(1,X ) 2 , then X satisfies the (DL)-condition 8 . From the inequality C −∞ (X ) C Z (X ) C N J (X ), Corollary 7 is better than Zhang's result 8 (Corollary 2.2), which also improves the result of Zhang and Cui, that if C NJ (X ) < 1 2 1 + 1 R(1,X ) 2 , then X satisfies the (DL)-condition 7 .…”

Herman's Approach to Quasi-Periodic Perturbations in the Reversible KAM Context 2

“…For instance, it was applied to coisotropic [20,25,26] and atropic [26] invariant tori of Hamiltonian [20] and locally Hamiltonian [25,26] systems. In [19,46,47,49], Herman's approach was employed in the case of systems with weak nondegeneracy conditions formulated in terms of the Brouwer topological degree. The papers [19,46,47] consider invariant tori in the reversible context 1 while the article [49] treats Hamiltonian systems depending quasi-periodically on time.…”

Persistence of Invariant Tori in Integrable Hamiltonian Systems Under Almost Periodic Perturbations

“…Xu and You [33] proved that if some frequency satisfies certain nonresonant condition and topological degree condition, the perturbed system still has an invariant torus with this frequency under Rüssmann's nondegeneracy condition. Zhang and Cheng [36] concerned with the persistence of invariant tori for nearly integrable Hamiltonian systems under time quasiperiodic perturbations, they proved that if the frequency of unperturbed system satisfies the Rüssmann's nondegeneracy condition and has nonzero Brouwer's topological degree at some Diophantine frequency, then invariant torus with frequency (Diophantine frequency and frequency of time quasi-periodic perturbation) satisfying the Diophantine condition persists under time quasi-periodic perturbations.…”