Unbounded Energy Growth in Hamiltonian Systems with a Slowly Varying Parameter

  • Vassili GelfreichEmail author
  • Dmitry Turaev


We study Hamiltonian systems which depend slowly on time. We show that if the corresponding frozen system has a uniformly hyperbolic invariant set with chaotic behaviour, then the full system has orbits with unbounded energy growth (under very mild genericity assumptions). We also provide formulas for the calculation of the rate of the fastest energy growth. We apply our general theory to non-autonomous perturbations of geodesic flows and Hamiltonian systems with billiard-like and homogeneous potentials. In these examples, we show the existence of orbits with the rates of energy growth that range, depending on the type of perturbation, from linear to exponential in time. Our theory also applies to non-Hamiltonian systems with a first integral.


Periodic Orbit Hamiltonian System Invariant Manifold Heteroclinic Cycle Energy Growth 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUnited Kingdom
  2. 2.Department of MathematicsBen Gurion University of the NegevBe’er ShevaIsrael

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