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Localized asymptotic behavior for almost additive potentials
Instability for a priori unstable Hamiltonian systems: A dynamical approach
1. | IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil, 22460-320, Brazil |
2. | IMJ, 4 place Jussieu, 75252 Paris Cedex, France |
References:
[1] |
V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations,, Russ. Math. Surv., 18 (1963), 9.
doi: 10.1070/RM1963v018n05ABEH004130. |
[2] |
V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom,, Dokl. Akad. Nauk SSSR, 156 (1964), 9.
|
[3] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,", [Dynamical Systems III], 3 (2006).
|
[4] |
M. Berti, L. Biasco and P. Bolle, Drift in phase space: A new variational mechanism with optimal diffusion time,, J. Math. Pures Appl., 82 (2003), 613.
doi: 10.1016/S0021-7824(03)00032-1. |
[5] |
P. Bernard, Perturbation of a partially hyperbolic Hamiltonian system, [Perturbation d'un hamiltonien partiellement hyperbolique],, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 189.
|
[6] |
P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems,, J. Amer. Math. Soc., 21 (2008).
doi: 10.1090/S0894-0347-08-00591-2. |
[7] |
U. Bessi, An approach to Arnol'd's diffusion through the calculus of variations,, Nonlinear Anal., 26 (1996), 1115.
doi: 10.1016/0362-546X(94)00270-R. |
[8] |
A. Bounemoura and J.-P. Marco, Improved exponential stability for near-integrable quasi-convex Hamiltonians,, Nonlinearity, 24 (2011), 97.
doi: 10.1088/0951-7715/24/1/005. |
[9] |
A. Bounemoura, Effective stability for Gevrey and finitely differentiable prevalent Hamiltonians,, to appear in Communication in Mathematical Physics, (2011). Google Scholar |
[10] |
A. Bounemoura, Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians,, Journal of Differential Equations, 249 (2010), 2905.
doi: 10.1016/j.jde.2010.06.004. |
[11] |
L. Chierchia and G. Gallavotti, Drift and diffusion in phase space,, Ann. Inst. Henri Poincaré, 60 (1994).
|
[12] |
J. Cresson and C. Guillet, Periodic orbits and Arnold diffusion,, Discrete Contin. Dyn. Syst., 9 (2003), 451.
|
[13] |
M. Chaperon, Stable manifolds and the Perron-Irwin method,, Ergodic Theory Dynam. Systems, 24 (2004), 1359.
doi: 10.1017/S0143385703000701. |
[14] |
C.-Q. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems,, Journal of Differential Geometry, 67 (2004), 457.
|
[15] |
C.-Q. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case,, Journal of Differential Geometry, 82 (2009), 229.
|
[16] |
A. Delshams, R. de la Llave and T. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: Heuristics and rigorous verification on a model,, Mem. Amer. Math. Soc., 179 (2006).
|
[17] |
M. R. Herman, Some open problems in dynamical systems,, in, 1998 (1998), 797.
|
[18] |
M. Chaperon, The Lipschitzian core of some invariant manifold theorems,, Ergodic Theory Dynam. Systems, 28 (2008), 1419.
doi: 10.1017/S0143385707000910. |
[19] |
M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (1977).
|
[20] |
V. Kaloshin, M. Levi and M. Saprykina, An example of nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension,, preprint, (2010). Google Scholar |
[21] |
A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function,, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527.
|
[22] |
V. Kaloshin, K. Zhang and Y. Zheng, "Almost Dense Orbit on Energy Surface,", Proceedings of the XVI-th ICMP, (2009), 314. Google Scholar |
[23] |
P. Lochak and J.-P. Marco, Diffusion times and stability exponents for nearly integrable analytic systems,, Central European Journal of Mathematics, 3 (2005), 342.
doi: 10.2478/BF02475913. |
[24] |
P. Lochak, J.-P. Marco and D. Sauzin, On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems,, Mem. Amer. Math. Soc., 163 (2003).
|
[25] |
P. Lochak, Arnold diffusion; a compendium of remarks and questions,, in, 533 (1995), 168.
|
[26] |
J.-P. Marco, Uniform lower bounds of the splitting for analytic symplectic systems,, preprint, (2005). Google Scholar |
[27] |
J.-P. Marco, Models for skew-products and polysystems,, C. R. Acad. Sci. Paris, 346 (2008), 203.
|
[28] |
J.-P. Marco, Arnold diffusion in a priori stable systems on $\A^3$,, in preparation, (2010). Google Scholar |
[29] |
J.-P. Marco, Generic properties of classical systems on the torus $\T^2$,, in preparation, (2010). Google Scholar |
[30] |
J.-P. Marco, Nets of hyperbolic annuli in generic nearly integrable systems on $\A^3$,, in preparation, (2010). Google Scholar |
[31] |
J.-P. Marco, Skew-products and polysystems in the neighborhood of hyperbolic annuli,, in preparation, (2010). Google Scholar |
[32] |
N. D. Mèzer, Arnol'd diffusion. I. Announcement of results,, J. Math. Sci. (N. Y.), 124 (2004), 5275.
doi: 10.1023/B:JOTH.0000047353.78307.09. |
[33] |
R. Moeckel, Generic drift on Cantor sets of annuli,, in, 292 (1999), 163.
|
[34] |
J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., II (1962), 1.
|
[35] |
J. Moser, "Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics,", Hermann Weyl Lectures, (1973).
|
[36] |
J.-P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems,,, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 199.
|
[37] |
J.-P. Marco and D. Sauzin, Wandering domains and random walks in Gevrey near-integrable systems,, Ergodic Theory Dynam. Systems, 24 (2004), 1619.
doi: 10.1017/S0143385703000786. |
[38] |
N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems,, Uspehi Mat. Nauk, 32 (1977).
|
[39] |
N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II,, Trudy Sem. Petrovsk., 5 (1979), 5.
|
[40] |
D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems,, Nonlinearity, 17 (2004), 1803.
doi: 10.1088/0951-7715/17/5/014. |
[41] |
J.-C. Yoccoz, Introduction to hyperbolic dynamics,, in, 464 (1993), 265.
|
[42] |
K. Zhang, Speed of Arnold diffusion for analytic Hamiltonian systems,, to appear in Inventiones Mathematicae, (2011). Google Scholar |
show all references
References:
[1] |
V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations,, Russ. Math. Surv., 18 (1963), 9.
doi: 10.1070/RM1963v018n05ABEH004130. |
[2] |
V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom,, Dokl. Akad. Nauk SSSR, 156 (1964), 9.
|
[3] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,", [Dynamical Systems III], 3 (2006).
|
[4] |
M. Berti, L. Biasco and P. Bolle, Drift in phase space: A new variational mechanism with optimal diffusion time,, J. Math. Pures Appl., 82 (2003), 613.
doi: 10.1016/S0021-7824(03)00032-1. |
[5] |
P. Bernard, Perturbation of a partially hyperbolic Hamiltonian system, [Perturbation d'un hamiltonien partiellement hyperbolique],, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 189.
|
[6] |
P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems,, J. Amer. Math. Soc., 21 (2008).
doi: 10.1090/S0894-0347-08-00591-2. |
[7] |
U. Bessi, An approach to Arnol'd's diffusion through the calculus of variations,, Nonlinear Anal., 26 (1996), 1115.
doi: 10.1016/0362-546X(94)00270-R. |
[8] |
A. Bounemoura and J.-P. Marco, Improved exponential stability for near-integrable quasi-convex Hamiltonians,, Nonlinearity, 24 (2011), 97.
doi: 10.1088/0951-7715/24/1/005. |
[9] |
A. Bounemoura, Effective stability for Gevrey and finitely differentiable prevalent Hamiltonians,, to appear in Communication in Mathematical Physics, (2011). Google Scholar |
[10] |
A. Bounemoura, Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians,, Journal of Differential Equations, 249 (2010), 2905.
doi: 10.1016/j.jde.2010.06.004. |
[11] |
L. Chierchia and G. Gallavotti, Drift and diffusion in phase space,, Ann. Inst. Henri Poincaré, 60 (1994).
|
[12] |
J. Cresson and C. Guillet, Periodic orbits and Arnold diffusion,, Discrete Contin. Dyn. Syst., 9 (2003), 451.
|
[13] |
M. Chaperon, Stable manifolds and the Perron-Irwin method,, Ergodic Theory Dynam. Systems, 24 (2004), 1359.
doi: 10.1017/S0143385703000701. |
[14] |
C.-Q. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems,, Journal of Differential Geometry, 67 (2004), 457.
|
[15] |
C.-Q. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case,, Journal of Differential Geometry, 82 (2009), 229.
|
[16] |
A. Delshams, R. de la Llave and T. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: Heuristics and rigorous verification on a model,, Mem. Amer. Math. Soc., 179 (2006).
|
[17] |
M. R. Herman, Some open problems in dynamical systems,, in, 1998 (1998), 797.
|
[18] |
M. Chaperon, The Lipschitzian core of some invariant manifold theorems,, Ergodic Theory Dynam. Systems, 28 (2008), 1419.
doi: 10.1017/S0143385707000910. |
[19] |
M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (1977).
|
[20] |
V. Kaloshin, M. Levi and M. Saprykina, An example of nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension,, preprint, (2010). Google Scholar |
[21] |
A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function,, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527.
|
[22] |
V. Kaloshin, K. Zhang and Y. Zheng, "Almost Dense Orbit on Energy Surface,", Proceedings of the XVI-th ICMP, (2009), 314. Google Scholar |
[23] |
P. Lochak and J.-P. Marco, Diffusion times and stability exponents for nearly integrable analytic systems,, Central European Journal of Mathematics, 3 (2005), 342.
doi: 10.2478/BF02475913. |
[24] |
P. Lochak, J.-P. Marco and D. Sauzin, On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems,, Mem. Amer. Math. Soc., 163 (2003).
|
[25] |
P. Lochak, Arnold diffusion; a compendium of remarks and questions,, in, 533 (1995), 168.
|
[26] |
J.-P. Marco, Uniform lower bounds of the splitting for analytic symplectic systems,, preprint, (2005). Google Scholar |
[27] |
J.-P. Marco, Models for skew-products and polysystems,, C. R. Acad. Sci. Paris, 346 (2008), 203.
|
[28] |
J.-P. Marco, Arnold diffusion in a priori stable systems on $\A^3$,, in preparation, (2010). Google Scholar |
[29] |
J.-P. Marco, Generic properties of classical systems on the torus $\T^2$,, in preparation, (2010). Google Scholar |
[30] |
J.-P. Marco, Nets of hyperbolic annuli in generic nearly integrable systems on $\A^3$,, in preparation, (2010). Google Scholar |
[31] |
J.-P. Marco, Skew-products and polysystems in the neighborhood of hyperbolic annuli,, in preparation, (2010). Google Scholar |
[32] |
N. D. Mèzer, Arnol'd diffusion. I. Announcement of results,, J. Math. Sci. (N. Y.), 124 (2004), 5275.
doi: 10.1023/B:JOTH.0000047353.78307.09. |
[33] |
R. Moeckel, Generic drift on Cantor sets of annuli,, in, 292 (1999), 163.
|
[34] |
J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., II (1962), 1.
|
[35] |
J. Moser, "Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics,", Hermann Weyl Lectures, (1973).
|
[36] |
J.-P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems,,, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 199.
|
[37] |
J.-P. Marco and D. Sauzin, Wandering domains and random walks in Gevrey near-integrable systems,, Ergodic Theory Dynam. Systems, 24 (2004), 1619.
doi: 10.1017/S0143385703000786. |
[38] |
N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems,, Uspehi Mat. Nauk, 32 (1977).
|
[39] |
N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II,, Trudy Sem. Petrovsk., 5 (1979), 5.
|
[40] |
D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems,, Nonlinearity, 17 (2004), 1803.
doi: 10.1088/0951-7715/17/5/014. |
[41] |
J.-C. Yoccoz, Introduction to hyperbolic dynamics,, in, 464 (1993), 265.
|
[42] |
K. Zhang, Speed of Arnold diffusion for analytic Hamiltonian systems,, to appear in Inventiones Mathematicae, (2011). Google Scholar |
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