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Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

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Abstract

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rüssmann’s non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus, numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang’s previous ones (1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11671392).

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Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Zhaodong Ding

  2. HUA Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zaijiu Shang

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China

    Zaijiu Shang

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  1. Zhaodong Ding
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  2. Zaijiu Shang
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Correspondence to Zhaodong Ding.

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Ding, Z., Shang, Z. Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems. Sci. China Math. 61, 1567–1588 (2018). https://doi.org/10.1007/s11425-018-9311-7

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  • DOI: https://doi.org/10.1007/s11425-018-9311-7

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