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The Isoenergetic KAM-Type Theorem at Resonant Case for Nearly Integrable Hamiltonian Systems
Weichao Qian,Yong Li,Xue Yang
Keywords:isoenergetic KAM-type theorem; resonant case; nearly integrable Hamiltonian system.
      In this paper, we study the persistence of resonant invariant tori on energy surfaces for a nearly integrable Hamiltonian system under the usual Russmann nondegenerate condition. By a quasilinear iterative scheme, we prove the following: (1) The majority of resonant tori on a given energy surface will persist under the Russmann nondegenerate condition. (2) The maximal number of the preserved frequency components of a perturbed torus is characterized by the loss between the maximal rank of the Hessian matrices from the unperturbed system and the nondegeneracy of resonance. (3) If the unperturbed system admits a subisoenergetic nondegeneracy on an energy surface, then the majority of the unperturbed resonant tori on the energy surface will persist and give rise to a family of perturbed tori of the same energy, whose frequency ratios of the respective “nondegenerate” components are preserved.