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Symplectic Geometric Algorithms for Hamiltonian Systems pp 549–580Cite as

KAM Theorem of Symplectic Algorithms

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Abstract

Numerical results have shown the overwhelming superiority of symplectic algorithms over the conventional non-symplectic systems, especially in simulating the global and structural dynamic behavior of the Hamiltonian systems. In the class of Hamiltonian systems, the most important and better-understood systems are completely integrable ones. Completely integrable systems exhibit regular dynamic behavior which corresponds to periodic and quasi-periodic motions in the phase spaces. In this chapter, we study problems as to whether and to what extent symplectic algorithms can simulate qualitatively and approximate quantitatively the periodic and quasi-periodic phase curves of integrable Hamiltonian systems.

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Authors and Affiliations

  1. Institute of Computational Mathematics and Scientific/Engineering Computing, Beijing, 100190, China

    Kang Feng & Mengzhao Qin

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  1. Kang Feng
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  2. Mengzhao Qin
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© 2010 Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg

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Feng, K., Qin, M. (2010). KAM Theorem of Symplectic Algorithms. In: Symplectic Geometric Algorithms for Hamiltonian Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01777-3_14

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