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On the Destruction of Resonant Lagrangean Tori in Hamiltonian Systems

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Recent Trends in Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 35))

Abstract

Starting from Poincaré’s fundamental problem of dynamics, we consider perturbations of integrable Hamiltonian systems in the neighbourhood of resonant Lagrangean (i.e. maximal) invariant tori with a single (internal) resonance. Applying KAM Theory and Singularity Theory we investigate how such a torus disintegrates when the action variables vary in the resonant surface. For open subsets of this surface the resulting lower dimensional tori are either hyperbolic or elliptic. For a better understanding of the dynamics, both qualitatively and quantitatively, we also investigate the singular tori and the way in which they are being unfolded by the action variables. In fact, if N is the number of degrees of freedom, singularities up to co-dimension N − 1 cannot be avoided. In the case of Kolmogorov non-degeneracy the singular tori are parabolic, while under the weaker non-degeneracy condition of Rüssmann the lower dimensional tori may also undergo e.g. umbilical bifurcations. We emphasize that this application of Singularity Theory only uses internal (or distinguished) parameters and no external ones.

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Notes

  1. 1.

    Any non-degenerate example of a co-dimension 1 bifurcation can be reduced to this case. Here we exclude a setting with symmetry or other structural restrictions [20].

  2. 2.

    Here we depart from the general theory of planar singularities that allows to transparently treat the relative equilibria. Indeed, the coordinates p and q already have a “meaning”, so we had to sharpen the usual assumption that the (homogeneous) 3-jet does not have multiple roots.

  3. 3.

    Here we use that the vector (1, e, …, e4) is Diophantine as well.

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Acknowledgements

We thank an anonymous referee for challenging us with the example\(H(\varphi,I) = H_{0}(I) +\epsilon H_{1}(\varphi )\) of a non-versal perturbation.

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Authors and Affiliations

  1. Johann Bernoulli Instituut, Rijksuniversiteit Groningen, 9747 AG, Groningen, The Netherlands

    Henk W. Broer

  2. Mathematisch Instituut, Universiteit Utrecht, 3508 TA, Utrecht, The Netherlands

    Heinz Hanßmann

  3. Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China

    Jiangong You

Authors
  1. Henk W. Broer
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  2. Heinz Hanßmann
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  3. Jiangong You
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Corresponding authors

Correspondence to Henk W. Broer , Heinz Hanßmann or Jiangong You .

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Editors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Andreas Johann

  2. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Hans-Peter Kruse

  3. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Florian Rupp

  4. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Stephan Schmitz

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Broer, H.W., Hanßmann, H., You, J. (2013). On the Destruction of Resonant Lagrangean Tori in Hamiltonian Systems. In: Johann, A., Kruse, HP., Rupp, F., Schmitz, S. (eds) Recent Trends in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 35. Springer, Basel. https://doi.org/10.1007/978-3-0348-0451-6_13

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