Repulsion from resonances  [ Répulsion par les résonances ] (2012)


Dolgopyat, Dmitry
Mémoires de la Société Mathématique de France, Tome 128 (2012) vi-119 p
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doi : 10.24033/msmf.439
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Bibliographie

[1] V. S. Afraimovich & L. P. Shilnikov« The ring principle in problems of interaction between two self-oscillating systems (Russian) », J. Appl. Math. Mech. 41 (1977), p. 618–627. MR 506941

[2] D. V. Anosov« Averaging in systems of ordinary differential equations with rapidly oscillating solutions (Russian) », Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), p. 721–742. MR 126592

[3] V. I. Arnold« Small denominators and problems of stability of motion in classical and celestial mechanics », Russian Math. Surv. 18 (1963), p. 85–191. MR 170705

[4] —, « Applicability conditions and an error bound for the averaging method for systems in the process of evolution through a resonance », Soviet Math. Doklady 6 (1965), p. 331–334. Zbl 0143.12001

[5] V. I. Arnold, V. V. Kozlov & A. I. Neishtadt« Mathematical aspects of classical and celestial mechanics », Encyclopaedia Math. Sci., vol. 3, Springer, Berlin, 3d éd., 2006. Zbl 1105.70002 | MR 2269239

[6] V. I. Bakhtin« Averaging in a general-position single-frequency system », Diff. Eq. 27 (1991), p. 1051–1061. MR 1140545

[7] —, « Cramer asymptotics in the averaging method for systems with fast hyperbolic motions (Russian) », Tr. Mat. Inst. Steklova 244 (2004), p. 65–86.

[8] N. N. Bogoliubov & Y. A. Mitropolsky« Asymptotic methods in the theory of non-linear oscillations », International Monographs on Advanced Math., Phys., Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961. MR 141845

[9] N. N. Bogolyubov & D. N. Zubarev« Method of asymptotic approximation for systems with rotating phase and its application to motion of charged particles in a magnetic field » (Russian), Ukr. Mat. Zh. 7 (1955), p. 5–17.

[10] M. Brin & M. I. Freidlin« On stochastic behavior of perturbed Hamiltonian systems », Erg. Th. Dyn. Sys. 20 (2000), p. 55–76. Zbl 0997.37020 | MR 1747030

[11] J. R. Cary, D. F. Escande & J. L. Tennyson« Adiabatic invariant change due to separatrix crossing », Phys. Rev. A 34 (1986), p. 4256–4275.

[12] J. R. Cary & R. T. Skodje« Phase change between separatrix crossings », Physica D 36 (1989), p. 287–316. Zbl 0687.70017 | MR 1012712

[13] N. Chernov & D. Dolgopyat« Brownian Motion I », Memoirs Amer. Math. Soc., vol. 198, 2009. Zbl 1173.60003 | MR 2499824

[14] B. V. Chirikov« The passage of a nonlinear oscillating system through resonance », Soviet Physics Doklady 4 (1959), p. 390. Zbl 0102.39705 | MR 108007

[15] J. De Simoi« Abundance of escaping orbits in a family of antiintegrable limits of the standard map », Thèse, University of Maryland, 2009. MR 2718075

[16] D. Dolgopyat« Limit theorems for partially hyperbolic systems », Trans. Amer. Math. Soc. 356 (2004), p. 1637–1689. Zbl 1031.37031 | MR 2034323

[17] —, « On differentiability of SRB states for partially hyperbolic systems », Invent. Math. 155 (2004), p. 389–449. Zbl 1059.37021 | MR 2031432

[18] —, « Averaging and invariant measures », Mosc. Math. J. 5 (2005), p. 537–576. Zbl 05140621 | MR 2241812

[19] —, « Bouncing balls in non-linear potentials », Discrete Contin. Dyn. Syst. 22 (2008), p. 165–182. Zbl 1154.37329 | MR 2410953

[20] P. Fatou« Sur le mouvement d’un système soumis à des forces de courte période », Bull Soc. Math. France 56 (1928), p. 98–139. Zbl 54.0834.01 | | MR 1504928

[21] M. I. Fredlin« The averaging principle and theorems on large deviations », Russian Math. Surv. 33 (1978), p. 117–176. MR 511884

[22] —, « Autonomous stochastic perturbations of dynamical systems », Acta Appl. Math. 78 (2003), p. 121–128. Zbl 1043.34064 | MR 2021775

[23] M. I. Freidlin« Random and deterministic perturbations of nonlinear oscillators », in Proc. ICM-98, Doc. Math., vol. Extra Vol. III, 1998, p. 223–235. Zbl 0908.60051 | | MR 1648157

[24] M. I. Freidlin & A. D. WentzellRandom perturbations of dynamical systems, 2nd éd., Grundlehren Math. Wiss., vol. 260, Springer, New York, 1998. MR 1652127

[25] I. I. Gihman & A. V. SkorohodThe theory of stochastic processes I, Grundlehren Math. Wiss., vol. 210, Springer, New York, 1974. MR 346882

[26] P. Goldreich« An explanation of the frequent occurrence of commensurable mean motions in the solar system », Monthly Notices Royal Astronomical Soc. 130 (1965), p. 159–181.

[27] P. Goldreich & S. Peale« Spin-orbit coupling in the solar system », Astronomical J. 71 (1966), p. 425–437.

[28] J. Guckenheimer, M. Wechselberger & L.-S. Young« Chaotic attractors of relaxation oscillators », Nonlinearity 19 (2006), p. 701–720. Zbl 1102.34028 | MR 2209295

[29] J. Henrard & A. Lemaitre« A mechanism of formation for the Kirkwood gaps », Icarus 55 (1983), p. 482–494.

[30] M. W. Hirsch, C. C. Pugh & M. ShubInvariant manifolds, Lect. Notes in Math., vol. 583, Springer, 1977. Zbl 0355.58009 | MR 501173

[31] A. P. Itin, A. I. Neishtadt & A. A. Vasiliev« Captures into resonance and scattering on resonance in dynamics of a charged relativistic particle in magnetic field and electrostatic wave », Phys. D 141 (2000), p. 281–296. Zbl 0982.78004 | MR 1761000

[32] T. Kasuga« On the adiabatic theorem for the Hamiltonian system of differential equations in the classical mechanics I–III », Proc. Japan Acad. Sci. 37 (1961), p. 366–382. Zbl 0114.14903 | MR 154458

[33] Y. Kifer« Averaging in dynamical systems and large deviations », Invent. Math. 110 (1992), p. 337–370. Zbl 0791.58072 | | MR 1185587

[34] —, « Modern dynamical systems and applications », in Some recent advances in averaging, Cambridge Univ. Press, Cambridge, 2004, p. 385–403.

[35] A. N. Kolmogorov« On conservation of conditionally periodic motions for a small change in Hamilton’s function », Dokl. Akad. Nauk SSSR (N.S.) (1954), p. 98 (Russian: 527–530). MR 68687

[36] M. Kruskal« Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic », J. Mathematical Phys. 3 (1962), p. 806–828. Zbl 0113.21201 | MR 151001

[37] P. Lochak & C. Meunier« Multiphase averaging for classical systems », Springer Appl. Math. Sci. 72 (1988). Zbl 0668.34044 | MR 959890

[38] L. I. Mandelstam & N. D. Papalexi« Uber die Begrundung einer Methode fur die Naherungslosung von Differentialgleichungen », J. Exp. Theor. Phys. 4 (1934), p. 117.

[39] J. Moser« Stable and random motions in dynamical systems. With special emphasis on celestial mechanics », Ann. Math. Studies 77. Zbl 0991.70002 | MR 442980

[40] A. I. Neishtadt« On resonant problems in nonlinear systems », Thèse, Moscow State University, 1975.

[41] —, « Passage through a resonance in the two-frequency problem », Sov. Phys. Dokl. 20 (1975), p. 189–191. Zbl 0325.70014

[42] —, « Passage through a separatrix in a resonance problem with a slowly-varying parameter », J. Appl. Math. Mech. 39 (1975), p. 594–605. Zbl 0356.70020

[43] —, « Averaging in multifrequency systems, II », Sov. Phys., Dokl. 21 (1976), p. 80–82.

[44] —, « Change of an adiabatic invariant at a separatrix », Sov. J. Plasma Phys. 12 (1986), p. 568–573.

[45] —, « On the change in the adiabatic invariant on crossing a separatrix in systems with two degrees of freedom », J. Appl. Math. Mech. 51 (1987), p. 586–592. Zbl 0677.70024

[46] —, « On perturbation theory of nonlinear resonant systems », 1988, Habilitation, Moscow State University.

[47] —, « Averaging, capture into resonances, and chaos in nonlinear systems », in Chaos, Amer. Inst. Phys., New York, 1990, p. 261–273.

[48] —, « Averaging and passage through resonances », in Proc. Int. Congress Math. (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, p. 1271–1283. Zbl 0780.34025

[49] —, « On destruction of adiabatic invariants in multi-frequency systems », in Equadiff 91, International Conference on Differential Equations, vol. 1, 1993, p. 195–207. Zbl 0938.34529

[50] —, « On probabilistic phenomena in perturbed systems », Selecta Math. Soviet. 12 (1993), p. 195–210. Zbl 0799.34048 | MR 1244836

[51] —, « On adiabatic invariance in two-frequency systems », in Hamiltonian systems with three or more degrees of freedom, Proceedings of NATO ASI, Series C, vol. 533, Kluwer, Dordrecht, p. 193–212. Zbl 0970.70016

[52] A. I. Neishtadt, V. V. Sidorenko & D. V. Treschev« Stable periodic motions in the problem of passage through a separatrix », Chaos 7 (1997), p. 2–11. Zbl 1002.34029 | MR 1439802

[53] A. I. Neishtadt & A. A. Vasiliev« Phase change between separatrix crossings in slow-fast Hamiltonian systems », Nonlinearity 18 (2005), p. 1393–1406. Zbl 1080.37064 | MR 2134900

[54] I. M. Operchuk« Study of statistical properties of multiple resonance passages », Thèse, Moscow State University, 2003.

[55] W. Ott & Q. Wang« Dissipative homoclinic loops and rank one chaos », preprint, arXiv:0802.4283.

[56] M. M. Peixoto« Structural stability on two-dimensional manifolds », Topology 1 (1962), p. 101–120. Zbl 0107.07103 | MR 142859

[57] O. Piro & M. Feingold« Diffusion in three-dimensional Liouvillian maps », Phys. Rev. Lett. 61 (1988), p. 1799–1802. MR 962603

[58] J. A. Sanders, F. Verhulst & J. MurdockAveraging methods in nonlinear dynamical systems, 2nd éd., Appl. Math. Sci., vol. 59, Springer, New York, 2007. Zbl 1128.34001 | MR 2316999

[59] D. L. Vainchtein, A. I. Neishtadt & I. Mezic« On passage through resonances in volume-preserving systems », Chaos 16 (2006), paper 043123. Zbl 1146.37350 | MR 2289293

[60] D. L. Vainchtein, E. V. Rovinsky, L. M. Zelenyi & A. I. Neishtadt« Resonances and particle stochastization in nonhomogeneous electromagnetic fields », J. Nonlinear Sci. 14 (2004), p. 173–205. Zbl 1086.78002 | MR 2041430

[61] D. L. Vainchtein, A. A. Vasiliev & A. I. Neishtadt« Electron dynamics in a parabolic magnetic field in the presence of an electrostatic wave », Plasma Physics Reports 35 (2009), p. 1021–1031.

[62] A. A. Vasilev, G. M. Zaslavskii, M. Y. Natenzon, A. I. Neishtadt, B. A. Petrovichev, R. Z. Sagdeev & A. A. Chernikov« Attractors and stochastic attractors of motion in a magnetic field », Soviet Phys. JETP 67 (1988), p. 2053–2062. MR 997935

[63] Q. Wang & L.-S. Young« From invariant curves to strange attractors », Comm. Math. Phys. 225 (2002), p. 275–304. Zbl 1080.37550 | MR 1889226