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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2020, Volume 310, Pages 19–32
DOI: https://doi.org/10.4213/tm4119
(Mi tm4119)
 

This article is cited in 1 scientific paper (total in 1 paper)

Local Adiabatic Invariants Near a Homoclinic Set of a Slow–Fast Hamiltonian System

Sergey V. Bolotin

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
Full-text PDF (232 kB) Citations (1)
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Abstract: In slow–fast systems, fast variables change at a rate of the order of one, and slow variables, at a rate of the order of $\varepsilon \ll 1$. The system obtained for $\varepsilon =0$ is said to be frozen. If the frozen (fast) system has one degree of freedom, then in the region where the level curves of the frozen Hamiltonian are closed there exists an adiabatic invariant. A. Neishtadt showed that near a separatrix of the frozen system the adiabatic invariant exhibits quasirandom jumps of order $\varepsilon $. In this paper we partially extend Neishtadt's result to the multidimensional case. We show that if the frozen system has a hyperbolic critical point possessing several transverse homoclinics, then for small $\varepsilon $ there exist trajectories shadowing homoclinic chains. The slow variables evolve in a quasirandom way, shadowing trajectories of systems with Hamiltonians similar to adiabatic invariants. This paper extends the work of V. Gelfreich and D. Turaev, who considered similar phenomena away from critical points of the frozen Hamiltonian.
Funding agency Grant number
Russian Science Foundation 19-71-30012
This work is supported by the Russian Science Foundation under grant 19-71-30012.
Received: January 8, 2020
Revised: April 25, 2020
Accepted: April 28, 2020
English version:
Proceedings of the Steklov Institute of Mathematics, 2020, Volume 310, Pages 12–24
DOI: https://doi.org/10.1134/S0081543820050028
Bibliographic databases:
Document Type: Article
UDC: 517.928.7+517.933
Language: Russian
Citation: Sergey V. Bolotin, “Local Adiabatic Invariants Near a Homoclinic Set of a Slow–Fast Hamiltonian System”, Selected issues of mathematics and mechanics, Collected papers. On the occasion of the 70th birthday of Academician Valery Vasil'evich Kozlov, Trudy Mat. Inst. Steklova, 310, Steklov Math. Inst., Moscow, 2020, 19–32; Proc. Steklov Inst. Math., 310 (2020), 12–24
Citation in format AMSBIB
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\by Sergey~V.~Bolotin
\paper Local Adiabatic Invariants Near a Homoclinic Set of a Slow--Fast Hamiltonian System
\inbook Selected issues of mathematics and mechanics
\bookinfo Collected papers. On the occasion of the 70th birthday of Academician Valery Vasil'evich Kozlov
\serial Trudy Mat. Inst. Steklova
\yr 2020
\vol 310
\pages 19--32
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4119}
\crossref{https://doi.org/10.4213/tm4119}
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\jour Proc. Steklov Inst. Math.
\yr 2020
\vol 310
\pages 12--24
\crossref{https://doi.org/10.1134/S0081543820050028}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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