Repulsion from resonances
Mémoires de la Société Mathématique de France, no. 128 (2012), 125 p.
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

We consider slow-fast systems with periodic fast motion and integrable slow motion in the presence of both weak and strong resonances. Assuming that the initial phases are random and that appropriate non-degeneracy assumptions are satisfied we prove that the effective evolution of the adiabatic invariants is given by a Markov process. This Markov process consists of the motion along the trajectories of a vector field with occasional jumps. The generator of the limiting process is computed from the dynamics of the system near strong resonances.

Nous considérons des systèmes « lents-rapides », dont le mouvement rapide est périodique et le mouvement lent intégrable, en présence de résonances faibles ou fortes. En supposant que les phases initiales sont aléatoires et que certaines conditions de non-dégénérescence sont satisfaites, nous démontrons que l’évolution effective des invariants adiabatiques est donnée par un processus de Markov. Ce processus de Markov consiste en un mouvement le long des trajectoires d’un champ de vecteurs qui peut présenter des sauts occasionnels. Le générateur du processus limite est calculé à partir de la dynamique du système au voisinage des résonances fortes.

DOI : https://doi.org/10.24033/msmf.439
Classification:  34C29,  70K65,  37D25,  60F17
Keywords: Averaging, Slow-Fast Systems, Markov Processes, Invariant Cones, Resonances
@book{MSMF_2012_2_128__1_0,
author = {Dolgopyat, Dmitry},
title = {Repulsion from resonances},
series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
publisher = {Soci\'et\'e math\'ematique de France},
number = {128},
year = {2012},
doi = {10.24033/msmf.439},
zbl = {1254.37027},
mrnumber = {2977412},
language = {en},
url = {http://www.numdam.org/item/MSMF_2012_2_128__1_0}
}

Dolgopyat, Dmitry. Repulsion from resonances. Mémoires de la Société Mathématique de France, Serie 2, , no. 128 (2012), 125 p. doi : 10.24033/msmf.439. http://www.numdam.org/item/MSMF_2012_2_128__1_0/

[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. | MR 2269239 | Zbl 1105.70002

[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. | MR 1747030 | Zbl 0997.37020

[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. | MR 1012712 | Zbl 0687.70017

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

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

[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. | MR 2034323 | Zbl 1031.37031

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

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

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

[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. | JFM 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. | MR 2021775 | Zbl 1043.34064

[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. | MR 1648157 | Zbl 0908.60051

[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. | MR 2209295 | Zbl 1102.34028

[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. | MR 501173 | Zbl 0355.58009

[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. | MR 1761000 | Zbl 0982.78004

[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. | MR 154458 | Zbl 0114.14903

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

[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. | MR 151001 | Zbl 0113.21201

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

[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. | MR 442980 | Zbl 0991.70002

[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. | MR 1244836 | Zbl 0799.34048

[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. | MR 1439802 | Zbl 1002.34029

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

[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. | MR 142859 | Zbl 0107.07103

[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. | MR 2316999 | Zbl 1128.34001

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

[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. | MR 2041430 | Zbl 1086.78002

[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. | MR 1889226 | Zbl 1080.37550