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Uspekhi Mat. Nauk, 2015, Volume 70, Issue 6(426), Pages 3–62 (Mi umn9692)  

This article is cited in 17 scientific papers (total in 17 papers)

The anti-integrable limit

S. V. Bolotin, D. V. Treschev

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: The anti-integrable limit is one of the convenient and relatively simple methods for the construction of chaotic hyperbolic invariant sets in Lagrangian, Hamiltonian, and other dynamical systems. This survey discusses the most natural context of the method, namely, discrete Lagrangian systems, and then presents examples and applications.
Bibliography: 75 titles.

Keywords: Lagrangian systems, Hamiltonian systems, chaos, hyperbolic sets, topological Markov chain, topological entropy.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


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English version:
Russian Mathematical Surveys, 2015, 70:6, 975–1030

Bibliographic databases:

UDC: 531.01
MSC: Primary 37D45; Secondary 37B10, 37B40
Received: 17.10.2015

Citation: S. V. Bolotin, D. V. Treschev, “The anti-integrable limit”, Uspekhi Mat. Nauk, 70:6(426) (2015), 3–62; Russian Math. Surveys, 70:6 (2015), 975–1030

Citation in format AMSBIB
\by S.~V.~Bolotin, D.~V.~Treschev
\paper The anti-integrable limit
\jour Uspekhi Mat. Nauk
\yr 2015
\vol 70
\issue 6(426)
\pages 3--62
\jour Russian Math. Surveys
\yr 2015
\vol 70
\issue 6
\pages 975--1030

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    This publication is cited in the following articles:
    1. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. V. V. Kozlov, D. V. Treschev, “Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics”, Sb. Math., 207:10 (2016), 1435–1449  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. S. V. Bolotin, “Degenerate billiards”, Proc. Steklov Inst. Math., 295 (2016), 45–62  mathnet  crossref  crossref  mathscinet  isi  elib
    4. M. N. Davletshin, D. V. Treschev, “Arnold diffusion in a neighborhood of strong resonances”, Proc. Steklov Inst. Math., 295 (2016), 63–94  mathnet  crossref  crossref  mathscinet  isi  elib
    5. Sergey V. Bolotin, “Degenerate Billiards in Celestial Mechanics”, Regul. Chaotic Dyn., 22:1 (2017), 27–53  mathnet  crossref  mathscinet
    6. S. V. Bolotin, V. V. Kozlov, “Topological approach to the generalized $n$-centre problem”, Russian Math. Surveys, 72:3 (2017), 451–478  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. S. V. Bolotin, V. V. Kozlov, “Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom”, Izv. Math., 81:4 (2017), 671–687  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. Alexey V. Ivanov, “Connecting Orbits near the Adiabatic Limit of Lagrangian Systems with Turning Points”, Regul. Chaotic Dyn., 22:5 (2017), 479–501  mathnet  crossref
    9. D. Treschev, “A locally integrable multi-dimensional billiard system”, Discrete Contin. Dyn. Syst., 37:10 (2017), 5271–5284  crossref  mathscinet  zmath  isi  scopus
    10. J. Féjoz, A. Knauf, R. Montgomery, “Lagrangian relations and linear point billiards”, Nonlinearity, 30:4 (2017), 1326–1355  crossref  mathscinet  zmath  isi  scopus
    11. S. V. Bolotin, “Jumps of energy near a separatrix in slow-fast Hamiltonian systems”, Russian Math. Surveys, 73:4 (2018), 725–727  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. V. V. Kozlov, D. V. Treschev, “Instability, asymptotic trajectories and dimension of the phase space”, Mosc. Math. J., 18:4 (2018), 681–692  mathnet  crossref  mathscinet
    13. 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  mathnet  crossref  mathscinet
    14. V. Schastnyy, D. Treschev, “On local integrability in billiard dynamics”, Exp. Math., 28:3 (2019), 362–368  crossref  isi
    15. R. Trevino, “Equilibrium configurations for generalized frenkel-kontorova models on quasicrystals”, Commun. Math. Phys., 371:1 (2019), 1–17  crossref  isi
    16. 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
    17. Portaluri A., Wu L., Yang R., “Linear Instability For Periodic Orbits of Non-Autonomous Lagrangian Systems”, Nonlinearity, 34:1 (2021), 237–272  crossref  zmath  isi
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