A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: Announcement of results
Authors:
Amadeu Delshams, Rafael de la Llave and Tere M. Seara
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 125-134
MSC (2000):
Primary 37J40; Secondary 70H08, 37D10, 70K70
DOI:
https://doi.org/10.1090/S1079-6762-03-00121-5
Published electronically:
December 4, 2003
MathSciNet review:
2029474
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Abstract: We present a geometric mechanism for diffusion in Hamiltonian systems. We also present tools that allow us to verify it in a concrete model. In particular, we verify it in a system which presents the large gap problem.
- V. I. Arnol′d and A. Avez, Ergodic problems of classical mechanics, W. A. Benjamin, Inc., New York-Amsterdam, 1968. Translated from the French by A. Avez. MR 0232910
- V. I. Arnol′d, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 91–192 (Russian). MR 0170705
- V. I. Arnol′d, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR 156 (1964), 9–12 (Russian). MR 0163026
- Massimiliano Berti and Philippe Bolle, A functional analysis approach to Arnold diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), no. 4, 395–450 (English, with English and French summaries). MR 1912262, DOI https://doi.org/10.1016/S0294-1449%2801%2900084-1
[BBB03]BertiBB03 M. Berti, L. Biasco, and P. Bolle. Drift in phase space: a new variational mechanism with optimal diffusion time. J. Math. Pures Appl. (9), 82(6):613–664, 2003.
- S. Bolotin and D. Treschev, Unbounded growth of energy in nonautonomous Hamiltonian systems, Nonlinearity 12 (1999), no. 2, 365–388. MR 1677779, DOI https://doi.org/10.1088/0951-7715/12/2/013
- L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 1, 144 (English, with English and French summaries). MR 1259103
[CG98]ChierchiaG98 L. Chierchia and G. Gallavotti. Erratum drift and diffusion in phase space. Ann. Inst. H. Poincaré Phys. Théor., 68:135, 1998.
- Boris V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52 (1979), no. 5, 264–379. MR 536429, DOI https://doi.org/10.1016/0370-1573%2879%2990023-1
- B. V. Chirikov, M. A. Lieberman, D. L. Shepelyansky, and F. M. Vivaldi, A theory of modulational diffusion, Phys. D 14 (1985), no. 3, 289–304. MR 793707, DOI https://doi.org/10.1016/0167-2789%2885%2990091-0
[CY03]ChengY03 Chong-Qing Cheng and Jun Yan. Existence of diffusion orbits in a priori unstable Hamiltonian systems, 2003. MP_ARC # 03-360.
- A. Delshams and P. Gutiérrez, Splitting potential and the Poincaré-Melnikov method for whiskered tori in Hamiltonian systems, J. Nonlinear Sci. 10 (2000), no. 4, 433–476. MR 1766491, DOI https://doi.org/10.1007/s003329910016
- Raphaël Douady and Patrice Le Calvez, Exemple de point fixe elliptique non topologiquement stable en dimension $4$, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 21, 895–898 (French, with English summary). MR 715330
- Amadeu Delshams, Rafael de la Llave, and Tere M. Seara, A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of ${\bf T}^2$, Comm. Math. Phys. 209 (2000), no. 2, 353–392. MR 1737988, DOI https://doi.org/10.1007/PL00020961
[DLS03a]DelshamsLS03 A. Delshams, R. de la Llave, and T. M. Seara. A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model. Preprint, 2003.
[DLS03b]DelshamsLS03a A. Delshams, R. de la Llave, and T. M. Seara. Orbits of unbounded energy in generic quasiperiodic perturbations of geodesic flows of certain manifolds. Preprint, 2003.
- Raphaël Douady, Stabilité ou instabilité des points fixes elliptiques, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 1, 1–46 (French). MR 944100
- Amadeu Delshams and Rafael Ramírez-Ros, Melnikov potential for exact symplectic maps, Comm. Math. Phys. 190 (1997), no. 1, 213–245. MR 1484553, DOI https://doi.org/10.1007/s002200050239
- Ernest Fontich and Pau Martín, Arnold diffusion in perturbations of analytic integrable Hamiltonian systems, Discrete Contin. Dynam. Systems 7 (2001), no. 1, 61–84. MR 1806373, DOI https://doi.org/10.3934/dcds.2001.7.61
- Ernest Fontich and Pau Martín, Hamiltonian systems with orbits covering densely submanifolds of small codimension, Nonlinear Anal. 52 (2003), no. 1, 315–327. MR 1938663, DOI https://doi.org/10.1016/S0362-546X%2802%2900115-3
- G. Gallavotti, Arnold’s diffusion in isochronous systems, Math. Phys. Anal. Geom. 1 (1998/99), no. 4, 295–312. MR 1692234, DOI https://doi.org/10.1023/A%3A1009893118532
- Philip J. Holmes and Jerrold E. Marsden, Melnikov’s method and Arnol′d diffusion for perturbations of integrable Hamiltonian systems, J. Math. Phys. 23 (1982), no. 4, 669–675. MR 649549, DOI https://doi.org/10.1063/1.525415
- Claude Wendell Horton Jr., Linda E. Reichl, and Victor G. Szebehely (eds.), Long-time prediction in dynamics, Nonequilibrium Problems in the Physical Sciences and Biology, vol. 2, John Wiley & Sons, Inc., New York, 1983. Papers from the Workshop on Long-Time Prediction in Nonlinear Conservative Dynamical Systems held in Lakeway, Tex., March 1981; A Wiley-Interscience Publication. MR 714714
[LW89]LlaveW89 R. de la Llave and C. E. Wayne. Whiskered and lower-dimensional tori in nearly integrable Hamiltonian systems. Preprint, 1989.
[Mat95]Mather95-96 J. N. Mather. Graduate course at Princeton, 95–96, and Lectures at Penn State, Spring 96, Paris, Summer 96, Austin, Fall 96, 1995.
[Mat02]Mather02 J. N. Mather. Arnold diffusion I: Announcement of results. Preprint, 2002.
- Richard Moeckel, Transition tori in the five-body problem, J. Differential Equations 129 (1996), no. 2, 290–314. MR 1404385, DOI https://doi.org/10.1006/jdeq.1996.0119
[Poi99]Poincare99 H. Poincaré. Les méthodes nouvelles de la mécanique céleste, volume 1, 2, 3. Gauthier-Villars, Paris, 1892–1899.
- Carles Simó (ed.), Hamiltonian systems with three or more degrees of freedom, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 533, Kluwer Academic Publishers Group, Dordrecht, 1999. MR 1720877
- Jeffrey Tennyson, Resonance transport in near-integrable systems with many degrees of freedom, Phys. D 5 (1982), no. 1, 123–135. MR 666532, DOI https://doi.org/10.1016/0167-2789%2882%2990054-9
- D. V. Treshchëv, A mechanism for the destruction of resonance tori in Hamiltonian systems, Mat. Sb. 180 (1989), no. 10, 1325–1346, 1439 (Russian); English transl., Math. USSR-Sb. 68 (1991), no. 1, 181–203. MR 1025685, DOI https://doi.org/10.1070/SM1991v068n01ABEH001371
[Tre03]Treschev03 D. Treschev. Evolution of slow variables in a priori unstable Hamiltonian systems, 2003.
- Zhihong Xia, Arnold diffusion: a variational construction, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 867–877. MR 1648133
[AA67]ArnoldA68 V. I. Arnold and A. Avez. Ergodic problems of classical mechanics. Benjamin, New York, 1967.
[Arn63]Arnold63 V. I. Arnold. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi Mat. Nauk, 18(6 (114)):91–192, 1963.
[Arn64]Arnold64 V. I. Arnold. Instability of dynamical systems with several degrees of freedom. Sov. Math. Doklady, 5:581–585, 1964.
[BB02]BertiB02 M. Berti and P. Bolle. A functional analysis approach to Arnold diffusion. Ann. Inst. H. Poincaré Anal. Non Linéaire, 19(4):395–450, 2002.
[BBB03]BertiBB03 M. Berti, L. Biasco, and P. Bolle. Drift in phase space: a new variational mechanism with optimal diffusion time. J. Math. Pures Appl. (9), 82(6):613–664, 2003.
[BT99]BolotinT99 S. Bolotin and D. Treschev. Unbounded growth of energy in nonautonomous Hamiltonian systems. Nonlinearity, 12(2):365–388, 1999.
[CG94]ChierchiaG94 L. Chierchia and G. Gallavotti. Drift and diffusion in phase space. Ann. Inst. H. Poincaré Phys. Théor., 60(1):144, 1994.
[CG98]ChierchiaG98 L. Chierchia and G. Gallavotti. Erratum drift and diffusion in phase space. Ann. Inst. H. Poincaré Phys. Théor., 68:135, 1998.
[Chi79]Chirikov79 B. V. Chirikov. A universal instability of many-dimensional oscillator systems. Phys. Rep., 52(5):264–379, 1979.
[CLSV85]ChirikovLSV85 B. V. Chirikov, M. A. Lieberman, D. L. Shepelyansky, and F. M. Vivaldi. A theory of modulational diffusion. Phys. D, 14(3):289–304, 1985.
[CY03]ChengY03 Chong-Qing Cheng and Jun Yan. Existence of diffusion orbits in a priori unstable Hamiltonian systems, 2003. MP_ARC # 03-360.
[DG00]DelshamsG00 A. Delshams and P. Gutiérrez. Splitting potential and the Poincaré-Melnikov method for whiskered tori in Hamiltonian systems. J. Nonlinear Sci., 10(4):433–476, 2000.
[DLC83]DouadyC83 Raphaël Douady and Patrice Le Calvez. Exemple de point fixe elliptique non topologiquement stable en dimension $4$. C. R. Acad. Sci. Paris Sér. I Math., 296(21):895–898, 1983.
[DLS00]DelshamsLS00 A. Delshams, R. de la Llave, and T. M. Seara. A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of $\mathbb {T}^ 2$. Comm. Math. Phys., 209(2):353–392, 2000.
[DLS03a]DelshamsLS03 A. Delshams, R. de la Llave, and T. M. Seara. A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model. Preprint, 2003.
[DLS03b]DelshamsLS03a A. Delshams, R. de la Llave, and T. M. Seara. Orbits of unbounded energy in generic quasiperiodic perturbations of geodesic flows of certain manifolds. Preprint, 2003.
[Dou88]Douady88 R. Douady. Stabilité ou instabilité des points fixes elliptiques. Ann. Sci. École Norm. Sup. (4), 21(1):1–46, 1988.
[DR97]DelshamsR97a A. Delshams and R. Ramírez-Ros. Melnikov potential for exact symplectic maps. Comm. Math. Phys., 190:213–245, 1997.
[FM01]FontichM01 Ernest Fontich and Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete Contin. Dynam. Systems, 7(1):61–84, 2001.
[FM03]FontichM03 Ernest Fontich and Pau Martín. Hamiltonian systems with orbits covering densely submanifolds of small codimension. Nonlinear Anal., 52(1):315–327, 2003.
[Gal99]Gallavotti98 G. Gallavotti. Arnold’s diffusion in isochronous systems. Math. Phys. Anal. Geom., 1(4):295–312, 1998/99.
[HM82]HolmesM82 P. J. Holmes and J. E. Marsden. Melnikov’s method and Arnol’d diffusion for perturbations of integrable Hamiltonian systems. J. Math. Phys., 23(4):669–675, 1982.
[HRS83]Lakeway Claude Wendell Horton, Jr., Linda E. Reichl, and Victor G. Szebehely, editors. Long-time prediction in dynamics, volume 2 of Nonequilibrium Problems in the Physical Sciences and Biology. John Wiley & Sons Inc., New York, 1983. Papers from the Workshop on Long-Time Prediction in Nonlinear Conservative Dynamical Systems held in Lakeway, Tex., March 1981, A Wiley-Interscience Publication.
[LW89]LlaveW89 R. de la Llave and C. E. Wayne. Whiskered and lower-dimensional tori in nearly integrable Hamiltonian systems. Preprint, 1989.
[Mat95]Mather95-96 J. N. Mather. Graduate course at Princeton, 95–96, and Lectures at Penn State, Spring 96, Paris, Summer 96, Austin, Fall 96, 1995.
[Mat02]Mather02 J. N. Mather. Arnold diffusion I: Announcement of results. Preprint, 2002.
[Moe96]Moeckel96 Richard Moeckel. Transition tori in the five-body problem. J. Differential Equations, 129(2):290–314, 1996.
[Poi99]Poincare99 H. Poincaré. Les méthodes nouvelles de la mécanique céleste, volume 1, 2, 3. Gauthier-Villars, Paris, 1892–1899.
[Sim99]Simo99 Carles Simó, editor. Hamiltonian systems with three or more degrees of freedom, Dordrecht, 1999. Kluwer Academic Publishers Group.
[Ten82]Tennyson82 Jeffrey Tennyson. Resonance transport in near-integrable systems with many degrees of freedom. Phys. D, 5(1):123–135, 1982.
[Tre89]Treschev89 D. V. Treshchëv. A mechanism for the destruction of resonance tori in Hamiltonian systems. Mat. Sb., 180(10):1325–1346, 1439, 1989.
[Tre03]Treschev03 D. Treschev. Evolution of slow variables in a priori unstable Hamiltonian systems, 2003.
[Xia98]Xia98 Zhihong Xia. Arnold diffusion: a variational construction. In Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Extra Vol. II, pages 867–877 (electronic), 1998.
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Additional Information
Amadeu Delshams
Affiliation:
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
Email:
Amadeu.Delshams@upc.es
Rafael de la Llave
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712-1802
Email:
llave@math.utexas.edu
Tere M. Seara
Affiliation:
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
Email:
tere.m-seara@upc.es
Keywords:
Nearly integrable Hamiltonian systems,
normal forms,
slow variables,
normally hyperbolic invariant manifolds,
KAM theory,
Arnold diffusion
Received by editor(s):
March 9, 2003
Received by editor(s) in revised form:
September 19, 2003
Published electronically:
December 4, 2003
Communicated by:
Svetlana Katok
Article copyright:
© Copyright 2003
American Mathematical Society