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Uspekhi Mat. Nauk, 2001, Volume 56, Issue 3(339), Pages 79–142 (Mi umn394)  

This article is cited in 46 scientific papers (total in 46 papers)

Splitting of separatrices: perturbation theory and exponential smallness

V. G. Gelfreicha, V. F. Lazutkin

a Saint-Petersburg State University

Abstract: This is a survey of the main results related to separatrix splitting for area-preserving maps and Hamiltonian systems with one and a half degrees of freedom. Special attention is paid to problems in which the separatrix splitting is exponentially small with respect to the perturbation parameter.

DOI: https://doi.org/10.4213/rm394

Full text: PDF file (890 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2001, 56:3, 499–558

Bibliographic databases:

UDC: 517.927+517.928+517.987
MSC: Primary 37J40, 37J45; Secondary 37D10, 37J10, 37C29, 37G20, 34E05, 34C37, 70K40, 7
Received: 23.05.2000

Citation: V. G. Gelfreich, V. F. Lazutkin, “Splitting of separatrices: perturbation theory and exponential smallness”, Uspekhi Mat. Nauk, 56:3(339) (2001), 79–142; Russian Math. Surveys, 56:3 (2001), 499–558

Citation in format AMSBIB
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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. Gelfreich V., “Near strongly resonant periodic orbits in a Hamiltonian system”, Proc. Natl. Acad. Sci. USA, 99:22 (2002), 13975–13979  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
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    6. J. Math. Sci. (N. Y.), 128:2 (2005), 2747–2753  mathnet  crossref  mathscinet  zmath
    7. Loskutov A., Janoe A., “Homoclinical chaos suppression”, International Conference Physics and Control, 1–4, 2003, 403–409  isi
    8. J. Math. Sci. (N. Y.), 128:2 (2005), 2726–2746  mathnet  crossref  mathscinet  zmath
    9. Vecheslavov V.V., “Chaos in a driven pendulum under asymmetric forcing”, J. Exp. Theor. Phys., 99:3 (2004), 663–668  crossref  adsnasa  isi  elib  scopus  scopus
    10. Loskutov A.Yu., Dzhanoev A.R., “Suppression of chaos in the vicinity of a separatrix”, J. Exp. Theor. Phys., 98:5 (2004), 1045–1053  crossref  adsnasa  isi  elib  scopus  scopus
    11. Gelfreich V., “Hamiltonian bifurcations and local analytic classification”, Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, Nato Science Series, Series II: Mathematics, Physics and Chemistry, 137, 2004, 251–265  crossref  mathscinet  isi
    12. Lerman L., Gelfreich V., “Slow-fast Hamiltonian dynamics near a ghost separatix loop”, J. Math. Sci. (N. Y.), 126:5 (2005), 1445–1466  crossref  crossref  mathscinet  zmath  elib  scopus
    13. Ramírez-Ros R., “Exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables”, Phys. D, 210:3-4 (2005), 149–179  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    14. Baldomá I., Fontich E., “Exponentially small splitting of separatrices in a weakly hyperbolic case”, J. Differential Equations, 210:1 (2005), 106–134  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    15. Viswanath D., “Stable manifolds and homoclinic points near resonances in the restricted three-body problem”, Celestial Mech. Dynam. Astronom., 94:2 (2006), 213–235  crossref  mathscinet  zmath  adsnasa  isi  scopus
    16. Ajisaka S., Tasaki Sh., “Reconnection of unstable/stable manifolds of the Harper map—asymptotics-beyond-all-orders approach”, Progr. Theoret. Phys., 116:4 (2006), 631–668  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    17. A. Yu. Loskutov, “Dynamical chaos: systems of classical mechanics”, Phys. Usp., 50:9 (2007), 939–964  mathnet  crossref  crossref  adsnasa  isi  elib
    18. G. N. Piftankin, D. V. Treschev, “Separatrix maps in Hamiltonian systems”, Russian Math. Surveys, 62:2 (2007), 219–322  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    19. Meiss J.D., “Visual explorations of dynamics: The standard map”, Pramana, 70:6 (2008), 965–988  crossref  adsnasa  isi  scopus
    20. N. E. Kulagin, L. M. Lerman, T. G. Shmakova, “On Radial Solutions of the Swift–Hohenberg Equation”, Proc. Steklov Inst. Math., 261 (2008), 183–203  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    21. Gelfreich V., Simó C., “High-precision computations of divergent asymptotic series and homoclinic phenomena”, Discrete Contin. Dyn. Syst. Ser. B, 10:2-3 (2008), 511–536  mathscinet  zmath  isi
    22. Simó C., Treschev D., “Stability islands in the vicinity of separatrices of near-integrable symplectic maps”, Discrete Contin. Dyn. Syst. Ser. B, 10:2-3 (2008), 681–698  mathscinet  zmath  isi  elib
    23. Simó C., Vieiro A., “Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps”, Nonlinearity, 22:5 (2009), 1191–1245  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    24. Gelfreich V., Gelfreikh N., “Unique resonant normal forms for area-preserving maps at an elliptic fixed point”, Nonlinearity, 22:4 (2009), 783–810  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    25. 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  scopus
    26. Proc. Steklov Inst. Math., 267 (2009), 76–90  mathnet  crossref  mathscinet  zmath  isi  elib
    27. Petrera M., Pfadler A., Suris Yu.B., “On integrability of Hirota-Kimura-type discretizations: experimental study of the discrete Clebsch system”, Experiment. Math., 18:2 (2009), 223–247  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    28. Bloor K., Luzzatto S., “Some remarks on the geometry of the standard map”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19:7 (2009), 2213–2232  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    29. Magnitskii N.A., “On the nature of dynamic chaos in a neighborhood of a separatrix of a conservative system”, Differ. Equ., 45:5 (2009), 662–669  crossref  mathscinet  zmath  isi  elib  elib  scopus
    30. 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
    31. Gelfreich V., Turaev D., “Universal dynamics in a neighborhood of a generic elliptic periodic point”, Regul. Chaotic Dyn., 15:2-3 (2010), 159–164  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    32. 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  scopus
    33. Babak G. Oskouei, Eva Kanso, Paul K. Newton, “Streamline bifurcations and scaling theory for a multiple-wake model”, International Journal of Non-Linear Mechanics, 2010  crossref  isi  scopus  scopus
    34. José Pedro Gaivão, Vassili Gelfreich, “Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift–Hohenberg equation as an example”, Nonlinearity, 24:3 (2011), 677  crossref  mathscinet  zmath  isi  scopus  scopus
    35. Simo C., Vieiro A., “Dynamics in chaotic zones of area preserving maps: Close to separatrix and global instability zones”, Physica D-Nonlinear Phenomena, 240:8 (2011), 732–753  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    36. Anton Gorodetski, “On Stochastic Sea of the Standard Map”, Commun. Math. Phys, 2011  crossref  mathscinet  isi  scopus  scopus
    37. Afendikov A., Fiedler B., Liebscher S., “Plane Kolmogorov flows and Takens-Bogdanov bifurcation without parameters: The singly reversible case”, Asymptot Anal, 72:1–2 (2011), 31–76  mathscinet  zmath  isi  elib
    38. Baldoma I., Martin P., “The Inner Equation for Generalized Standard Maps”, SIAM J. Appl. Dyn. Syst., 11:3 (2012), 1062–1097  crossref  mathscinet  zmath  isi
    39. Jacopo De Simoi, “On cyclicity-one elliptic islands of the standard map”, JMD, 7:2 (2013), 153  crossref  mathscinet  zmath  isi  scopus
    40. Vassili Gelfreich, Lev Lerman, “Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation”, Regul. Chaotic Dyn., 19:6 (2014), 635–655  mathnet  crossref  mathscinet  zmath
    41. Chardard F., Bridges T.J., “Transversality of Homoclinic Orbits, the Maslov Index and the Symplectic Evans Function”, Nonlinearity, 28:1 (2015), 77–102  crossref  mathscinet  zmath  isi  scopus  scopus
    42. Regina Martínez, Carles Simó, “Return Maps, Dynamical Consequences and Applications”, Qual. Theory Dyn. Syst, 2015  crossref  mathscinet  isi  scopus  scopus
    43. Sergey M. Saulin, Dmitry V. Treschev, “On the Inclusion of a Map Into a Flow”, Regul. Chaotic Dyn., 21:5 (2016), 538–547  mathnet  crossref  mathscinet  zmath  elib
    44. Martin P., Ramirez-Ros R., Tamarit-Sariol A., “Exponentially Small Asymptotic Formulas for the Length Spectrum in Some Billiard Tables”, Exp. Math., 25:4 (2016), 416–440  crossref  mathscinet  zmath  isi  elib  scopus
    45. Fontich E., Simo C., Vieiro A., “Splitting of the Separatrices After a Hamiltonian-Hopf Bifurcation Under Periodic Forcing”, Nonlinearity, 32:4 (2019), 1440–1493  crossref  mathscinet  zmath  isi  scopus
    46. Berger P., Turaev D., “On Herman'S Positive Entropy Conjecture”, Adv. Math., 349 (2019), 1234–1288  crossref  isi
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