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Regul. Chaotic Dyn., 2019, Volume 24, Issue 6, Pages 682–703 (Mi rcd1033)  

This article is cited in 2 scientific papers (total in 2 papers)

Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System

Sergey V. Bolotinab

a University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI 53706-1325, USA
b Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: We consider a Hamiltonian system depending on a parameter which slowly changes with rate $\varepsilon \ll 1$. If trajectories of the frozen autonomous system are periodic, then the system has adiabatic invariant which changes much slower than energy. For a system with 1 degree of freedom and a figure 8 separatrix, Anatoly Neishtadt [18] showed that for trajectories crossing the separatrix, the adiabatic invariant, and hence the energy, have quasirandom jumps of order $\varepsilon$. We prove a partial analog of Neishtadt's result for a system with $n$ degrees of freedom such that the frozen system has a hyperbolic equilibrium possessing several homoclinic orbits. We construct trajectories staying near the homoclinic set with energy having jumps of order $\varepsilon$ at time intervals of order $|\ln\varepsilon|$, so the energy may grow with rate $\varepsilon/|\ln\varepsilon|$. Away from the homoclinic set faster energy growth is possible: if the frozen system has chaotic behavior, Gelfreich and Turaev [16] constructed trajectories with energy growth rate of order $\varepsilon$.

Keywords: Hamiltonian system, homoclinic orbit, action functional, Poincare function, symplectic relation, separatrix map, adiabatic invariant

Funding Agency Grant Number
Russian Science Foundation 19-71-30012
The research was funded by a grant from the Russian Science Foundation (Project No. 19-71-30012).


DOI: https://doi.org/10.1134/S1560354719060078

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Bibliographic databases:

MSC: 37D, 37J, 70H
Received: 22.10.2019
Accepted:07.11.2019
Language:

Citation: Sergey V. Bolotin, “Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System”, Regul. Chaotic Dyn., 24:6 (2019), 682–703

Citation in format AMSBIB
\Bibitem{Bol19}
\by Sergey V. Bolotin
\paper Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 6
\pages 682--703
\mathnet{http://mi.mathnet.ru/rcd1033}
\crossref{https://doi.org/10.1134/S1560354719060078}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4040814}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85076348314}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Sergey V. Bolotin, “Local Adiabatic Invariants Near a Homoclinic Set of a Slow–Fast Hamiltonian System”, Proc. Steklov Inst. Math., 310 (2020), 12–24  mathnet  crossref  crossref  isi  elib
    2. S. V. Bolotin, “Crossing of the Critical Energy Level in Hamiltonian Systems with Slow Dependence on Time”, Math. Notes, 110:6 (2021), 956–959  mathnet  crossref
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