On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition

Dongfeng Zhang; Junxiang Xu

doi:10.3934/dcds.2006.16.635

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2006, Volume 16, Issue 3: 635-655. Doi: 10.3934/dcds.2006.16.635

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On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition

Dongfeng Zhang^1, and
Junxiang Xu^2,

1.
Department of Mathematics, Southeast University, Nanjing 210096
2.
Department of Mathematics, Southeast University, Nanjing 211189

Received: December 31, 2005

Revised: May 31, 2006

Published: August 2006

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Abstract

In this paper we prove the persistence of elliptic lower dimensional invariant tori for nearly integrable Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth with respect to parameters in the sense of Whitney, with a Gevrey index depending on the Gevrey class of Hamiltonian systems and on the exponent in the Diophantine condition. Moreover the Gevrey index should be optimal for the Diophantine condition in the proof of our theorem.

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Mathematics Subject Classification: 70H08, 37J40, 34C27.

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