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Uspekhi Mat. Nauk, 2007, Volume 62, Issue 2(374), Pages 3–108 (Mi umn6804)  

This article is cited in 26 scientific papers (total in 26 papers)

Separatrix maps in Hamiltonian systems

G. N. Piftankina, D. V. Treschevb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The separatrix map is constructed for some classes of problems in Hamiltonian dynamics. The formulae obtained are used to study two-dimensional symplectic maps close to integrable maps: elliptic periodic trajectories passing through separatrix lobes are constructed, and some estimates for the width of the stochastic layer are given. For Hamiltonian systems with two and a half degrees of freedom it is proved that the Arnol'd diffusion in the a priori unstable case is generic, and in the Mather problem trajectories are constructed for which the mean energy growth is linear in time.

DOI: https://doi.org/10.4213/rm6804

Full text: PDF file (1577 kB)
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English version:
Russian Mathematical Surveys, 2007, 62:2, 219–322

Bibliographic databases:

Document Type: Article
UDC: 531.01
MSC: Primary 37J40, 37J10; Secondary 34C37, 37J45, 37D10, 37D30, 37C55, 70H05
Received: 01.02.2007

Citation: G. N. Piftankin, D. V. Treschev, “Separatrix maps in Hamiltonian systems”, Uspekhi Mat. Nauk, 62:2(374) (2007), 3–108; Russian Math. Surveys, 62:2 (2007), 219–322

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Soskin S.M., Mannella R., Yevtushenko O.M., “Matching of separatrix map and resonant dynamics, with application to global chaos onset between separatrices”, Phys. Rev. E, 77:3 (2008), 036221, 29 pp.  crossref  mathscinet  adsnasa  isi  elib  scopus
    2. Delshams A., Huguet G., “Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems”, Nonlinearity, 22:8 (2009), 1997–2077  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Soskin S.M., Mannella R., “Maximal width of the separatrix chaotic layer”, Phys. Rev. E, 80:6 (2009), 066212, 17 pp.  crossref  adsnasa  isi  scopus
    4. Soskin S.M., Mannella R., Yevtushenko O.M., Filiasi M., “Acceleration of the chaotic and noise-induced transport in adiabatically driven spatially periodic systems”, Noise and fluctuations, AIP Conference Proceedings, 1129, 2009, 21–24  crossref  adsnasa  isi  scopus
    5. Soskin S.M., Mannella R., “New approach to the treatment of separatrix chaos”, Noise and fluctuations, AIP Conference Proceedings, 1129, 2009, 25–28  crossref  adsnasa  isi  scopus
    6. Simó C., Vieiro A., “Planar radial weakly dissipative diffeomorphisms”, Chaos, 20:4 (2010), 043138, 18 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. Soskin S.M., McClintock P.V.E., Fromhold T.M., Khovanov I.A., Mannella R., “Stochastic webs and quantum transport in superlattices: an introductory review”, Contemporary Physics, 51:3 (2010), 233–248  crossref  adsnasa  isi  elib  scopus
    8. Wang Qiudong, Oksasoglu A., “Dynamics of homoclinic tangles in periodically perturbed second-order equations”, J. Differential Equations, 250:2 (2011), 710–751  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. Simo C., Vieiro A., “Dynamics in chaotic zones of area preserving maps: Close to separatrix and global instability zones”, Phys. D, 240:8 (2011), 732–753  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    10. Delshams A., Huguet G., “A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems”, J. Differential Equations, 250:5 (2011), 2601–2623  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    11. Simó C., Vieiro A., “Some remarks on the abundance of stable periodic orbits inside homoclinic lobes”, Phys. D, 240:24 (2011), 1936–1953  crossref  mathscinet  zmath  isi  scopus
    12. Soskin S.M., Mannella R., Yevtushenko O.M., Khovanov I.A., McClintock P.V.E., “A new approach to the treatment of separatrix chaos”, Fluct. Noise Lett., 11:01 (2012), 1240002, 12 pp.  crossref  isi  elib  scopus
    13. Treschev D., “Arnold diffusion far from strong resonances in multidimensionala prioriunstable Hamiltonian systems”, Nonlinearity, 25:9 (2012), 2717–2757  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. O. M. Kiselev, “Oscillations near a separatrix in the Duffing equation”, Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 82–94  mathnet  crossref  isi  elib
    15. S V Gonchenko, C Simó, A Vieiro, “Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight”, Nonlinearity, 26:3 (2013), 621  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    16. Sergey Bolotin, Piero Negrini, “Shilnikov Lemma for a Nondegenerate Critical Manifold of a Hamiltonian System”, Regul. Chaotic Dyn., 18:6 (2013), 774–800  mathnet  crossref  mathscinet  zmath
    17. O.M. Kiselev, N. Tarkhanov, “The capture of a particle into resonance at potential hole with dissipative perturbation”, Chaos, Solitons & Fractals, 58 (2014), 27  crossref  mathscinet  zmath  adsnasa  isi  scopus
    18. S. V. Bolotin, D. V. Treschev, “The anti-integrable limit”, Russian Math. Surveys, 70:6 (2015), 975–1030  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    19. 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
    20. Delshams A. de la Llave R. Seara T.M., “Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion”, Adv. Math., 294 (2016), 689–755  crossref  mathscinet  zmath  isi  elib  scopus
    21. Guardia M., Kaloshin V., Zhang J., “A Second Order Expansion of the Separatrix Map for Trigonometric Perturbations of a Priori Unstable Systems”, Commun. Math. Phys., 348:1 (2016), 321–361  crossref  mathscinet  zmath  isi  elib  scopus
    22. Fejoz J. Guardia M. Kaloshin V. Roldan P., “Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem”, J. Eur. Math. Soc., 18:10 (2016), 2315–2403  crossref  mathscinet  zmath  isi  elib  scopus
    23. Gelfreich V. Turaev D., “Arnold Diffusion in a Priori Chaotic Symplectic Maps”, Commun. Math. Phys., 353:2 (2017), 507–547  crossref  mathscinet  zmath  isi  scopus
    24. Gidea M., de la Llave R., “Perturbations of Geodesic Flows By Recurrent Dynamics”, J. Eur. Math. Soc., 19:3 (2017), 905–956  crossref  mathscinet  zmath  isi  scopus
    25. Simo C., “Some Questions Looking For Answers in Dynamical Systems”, Discret. Contin. Dyn. Syst., 38:12, SI (2018), 6215–6239  crossref  mathscinet  isi  scopus
    26. Delshams A., Guillamon A., Huguet G., “Quasiperiodic Perturbations of Heteroclinic Attractor Networks”, Chaos, 28:10 (2018), 103111  crossref  mathscinet  zmath  isi  scopus
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