Unbounded energy growth in Hamiltonian systems with a slowly varying parameter

Vassili Gelfreich, Dmitry Turaev

Research output: Contribution to journalArticlepeer-review

57 Scopus citations


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.

Original languageEnglish
Pages (from-to)769-794
Number of pages26
JournalCommunications in Mathematical Physics
Issue number3
StatePublished - 1 Nov 2008

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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