THE EXISTENCE OF RESPONSE TORI FOR HAMILTONIAN WITH NORMAL DEGENERACY

Lu Xu; Wen Si; Mengmeng Wu

doi:10.11948/20230503

2024 Volume 14 Issue 6

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Lu Xu, Wen Si, Mengmeng Wu. THE EXISTENCE OF RESPONSE TORI FOR HAMILTONIAN WITH NORMAL DEGENERACY[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3227-3259. doi: 10.11948/20230503

Citation:

Lu Xu, Wen Si, Mengmeng Wu. THE EXISTENCE OF RESPONSE TORI FOR HAMILTONIAN WITH NORMAL DEGENERACY[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3227-3259. doi: 10.11948/20230503

THE EXISTENCE OF RESPONSE TORI FOR HAMILTONIAN WITH NORMAL DEGENERACY

1.
School of Mathematics, Jilin University, Qianjin Street No. 2699, 130012 Changchun, China
2.
School of Mathematics, Shandong University, Shanda Road, 250100 Jinan, China

Author Bio: Email: xulu@jlu.edu.cn(L. Xu); Email: a2833481657@163.com(M. Wu)
Corresponding author: Email: siwenmath@sdu.edu.cn(W. Si)
Fund Project: The first author was partially supported by National Natural Science Foundation of China (Grant No. 12271204) and the Department project of Science and Technology of Jilin Province (Grant No. 20200201265JC). The second author was partially supported by the National Natural Science Foundation of China (Grant No. 12001315, 11971261, 11571201, 12071255) and Shandong Provincial Natural Science Foundation, China (Grant No. ZR2020MA015)

Abstract

Abstract

In this paper, we prove the existence of response tori for a general Hamiltonian with normal degeneracy which will be shown as (1.1). When the perturbation is independent of action varible $y$, it can be seen as the energy function of several quasi-periodically forced oscillator equations (1.2). Most of the previous results focus on a single oscillator equation and prove the existence of response solutions under certain non-degenerate assumptions. In the present paper, we will consider high dimensional system (1.2) coupled by oscillator equations in different degenerate types. We will prove that the response solutions still exist around perturbed equilibria, which reveals the mechanics of the existence of response solution for a system coupled by degenerate nonlinear oscillator equations. For the sake of generality, we will actually consider a general Hamiltonian normal form and prove the persistence of invariant tori with fixed Diophantine frequency $\omega$ by the methods of finding relative equilibria, improving the order of perturbations, KAM iterations and measure estimates. The result can be applied to prove the existence of response solutions of the above system (1.2).
- Normal degeneracy /
- KAM theory /
- response solutions
MSC: 37J40, 70H08

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References

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