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Multiple-Time-Scale Dynamical Systems pp 1–27Cite as

Homoclinic Orbits to Invariant Tori in Hamiltonian Systems

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 122))

Abstract

We consider a perturbation of an integrable Hamiltonian system which possesses invariant tori with coincident whiskers (like some rotators and a pendulum). Our goal is to measure the splitting distance between the perturbed whiskers, putting emphasis on the detection of their intersections, which give rise to homoclinic orbits to the perturbed tori. A geometric method is presented which takes into account the Lagrangian properties of the whiskers. In this way, the splitting distance is the gradient of a splitting potential. In the regular case (also known as a priori-unstable: The Lyapunov exponents of the whiskered tori remain fixed), the splitting potential is well-approximated by a Melnikov potential. This method is designed as a first step in the study of the singular case (also known as a priori-stable: The Lyapunov exponents of the whiskered tori approach to zero when the perturbation tends to zero).

This work was supported in part by the EC grant ERBCHRXCT940460.

Also supported in part by the Spanish grant DGICYT PB94-0215 and the Catalan grant CIRIT 1996SGR-000105.

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Author information

Authors and Affiliations

  1. Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028, Barcelona, Spain

    Amadeu Delshams

  2. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, 08028, Barcelona, Spain

    Pere Gutiérrez

Authors
  1. Amadeu Delshams
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  2. Pere Gutiérrez
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Editors and Affiliations

  1. Division of Applied Mathematics and Lefschetz Center for Dynamical Systems, Brown University, Providence, RI, 02912, USA

    Christopher K. R. T. Jones

  2. United Technologies Research Center, East Hartford, CT, 06108, USA

    Alexander I. Khibnik

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Delshams, A., Gutiérrez, P. (2001). Homoclinic Orbits to Invariant Tori in Hamiltonian Systems. In: Jones, C.K.R.T., Khibnik, A.I. (eds) Multiple-Time-Scale Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 122. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0117-2_1

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  • DOI: https://doi.org/10.1007/978-1-4613-0117-2_1

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