THE ISOENERGETIC KAM-TYPE THEOREM AT RESONANT CASE fOR NEARLY INTEGRABLE HAMILTONIAN SYSTEMS

Weichao Qian; Yong Li; Xue Yang

doi:10.11948/20180008

2019 Volume 9 Issue 5

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Weichao Qian, Yong Li, Xue Yang. THE ISOENERGETIC KAM-TYPE THEOREM AT RESONANT CASE fOR NEARLY INTEGRABLE HAMILTONIAN SYSTEMS[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 1616-1638. doi: 10.11948/20180008

Citation:

Weichao Qian, Yong Li, Xue Yang. THE ISOENERGETIC KAM-TYPE THEOREM AT RESONANT CASE fOR NEARLY INTEGRABLE HAMILTONIAN SYSTEMS[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 1616-1638. doi: 10.11948/20180008

THE ISOENERGETIC KAM-TYPE THEOREM AT RESONANT CASE fOR NEARLY INTEGRABLE HAMILTONIAN SYSTEMS

1.
School of Mathematics and Statistics $\&$ Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, 5268 People Street, 130024, China
2.
School of Mathematics, Jilin University, 2699 Qianjin Street, 130012, China

Corresponding author: Email address:liyongmath@163.com(Y. Li)
Fund Project: The second author was supposed by National Basic Research Program of China (grant No. 2013CB834100), NSFC (grant No. 11571065), JilinDRC (grant No. 2017C028-1). The third author was supposed by Science and Technology Developing Plan of Jilin Province (grant No. 20180101220JC)

Abstract

Abstract

In this paper, we study the persistence of resonant invariant tori on energy surfaces for nearly integrable Hamiltonian systems under the usual R$\ddot{u}$ssmann nondegenerate condition. By a quasilinear iterative scheme, we prove the following things: (1) The majority of resonant tori on a given energy surface will be persisted under R$\ddot{u}$ssmann nondegenerate condition. (2) The maximal number about the preserved frequency components on a perturbed torus is characterized by the smaller of the maximal rank of the Hessian matrices of the unperturbed system and the nondegeneracy of resonance. (3) If unperturbed systems admit subisoenergetic nondegeneracy on an energy surface, then the majority of the unperturbed resonant tori on the energy surface will be persisted and give rise to a family of perturbed tori with the same energy, whose frequency ratios among respective "nondegenerate" components are preserved.
- Isoenergetic KAM-type theorem /
- resonant case /
- nearly integrable Hamiltonian systems
MSC: 37J40

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References

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