Whiskered Tori for Presymplectic Dynamical Systems

Abstract

We prove persistence result of whiskered tori for the dynamical system which preserves an exact presymplectic form. The results are given in an a-posteriori format. Given an approximate solution of an invariance equation which satisfies some non-degeneracy assumptions, we conclude that there is a true solution close by. The proof is based on certain iterative procedure by which the accuracy of the approximate solutions of the invariance equation can be improved. The iterative procedure is not based on transformation theory, which is cumbersome for presymplectic systems, but on finding corrections to the solutions of the invariance equation. This iterative procedure takes advantage of identities that come from the preservation of the geometric structure and leads to a very efficient numerical method which has low storage requirements, low operator count per step and it is quadratically convergent. We note that a particular case of presymplectic systems is symplectic perturbed by quasi-periodic systems.

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Notes

  1. 1.

    A easy example when the assumption A1) does not hold is a product, \(M= A \times {{\mathbb {T}}}^l\), A is a symplectic manifold. When \(f(a,b) = (g(a), h(b))\), \(g(0) = 0\) and g preserves the symplectic form in A. In this case, M could be a manifold with presymplectic form. Then \(0 \times {{\mathbb {T}}}^l\) is an invariant torus of the presymplectic form in M. Depending on the dynamics of h, the kernel may contain (un)stable directions.

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Correspondence to Lu Xu.

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R.L. has been partially supported by NSF Grant DMS-1800241. L.X. has been supported by NSFC Grant 11401251. The collaboration of the authors benefited from the hospitality of the JLU-GT Joint institute for Theoretical Science and the author L.X visited Georgia Institute of Technology during January to July, 2017 supported by the CSC and Jilin University. This material is also based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while both authors were in residence at the Mathematical Science Research Institute in Berkeley, California, during the Fall 2018 semester.

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de la Llave, R., Xu, L. Whiskered Tori for Presymplectic Dynamical Systems. J Dyn Diff Equat 33, 1–34 (2021). https://doi.org/10.1007/s10884-020-09823-w

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Keywords

  • Whiskered tori
  • KAM theory
  • Small divisors
  • Presymplectic system

Mathematics Subject Classification

  • Primary 37K55
  • 70K43
  • 58F05
  • 70H15