Instability for <em>a priori</em> unstable Hamiltonian systems: A dynamical approach

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March 2012, 32(3): 753-793. doi: 10.3934/dcds.2012.32.753

Instability for a priori unstable Hamiltonian systems: A dynamical approach

Abed Bounemoura ^1, and Edouard Pennamen ^2,

1.	IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil, 22460-320, Brazil
2.	IMJ, 4 place Jussieu, 75252 Paris Cedex, France

Received October 2010 Revised May 2011 Published October 2011

Abstract
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In this article, we consider an a priori unstable Hamiltonian system with three degrees of freedom, for which we construct a drifting solution with an optimal time of instability. Such a result has been already proved by Berti, Bolle and Biasco using variational arguments, and by Treschev using his separatrix map theory. Our approach is new: it is based on a special type of symbolic dynamics corresponding to the random iteration of a family of twist maps of the annulus, and it gives the first concrete application of this idea introduced by Moeckel in an abstract setting and further studied by Marco. Our method should also be useful in obtaining the optimal time of instability in the more difficult context of a priori stable Hamiltonian systems.

Keywords: Arnold diffusion., Hamiltonian systems, Perturbation theory.

Mathematics Subject Classification: Primary: 37J40; Secondary: 37D0.

Citation: Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753

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), 9-36. doi: 10.1070/RM1963v018n05ABEH004130.
[2]	V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12.
[3]	V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," [Dynamical Systems III], Transl. from the Russian original by E. Khukhro, Third edition, Encyclopedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 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), 613-664. 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), 189-195.
[6]	P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615–-669. 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), 1115-1135. 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), 97-112. 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.
[10]	A. Bounemoura, Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians, Journal of Differential Equations, 249 (2010), 2905-2920. doi: 10.1016/j.jde.2010.06.004.
[11]	L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. Henri Poincaré, Phys. Théor., 60 (1994), 144 pp.
[12]	J. Cresson and C. Guillet, Periodic orbits and Arnold diffusion, Discrete Contin. Dyn. Syst., 9 (2003), 451-470.
[13]	M. Chaperon, Stable manifolds and the Perron-Irwin method, Ergodic Theory Dynam. Systems, 24 (2004), 1359-1394. 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), 457-517.
[15]	C.-Q. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, Journal of Differential Geometry, 82 (2009), 229-277.
[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), viii+141 pp.
[17]	M. R. Herman, Some open problems in dynamical systems, in "Proceedings of the International Congress of Mathematicians," Vol. II (Berlin, 1998), Doc. Math., 1998, Extra Vol. II, 797-808.
[18]	M. Chaperon, The Lipschitzian core of some invariant manifold theorems, Ergodic Theory Dynam. Systems, 28 (2008), 1419-1441. doi: 10.1017/S0143385707000910.
[19]	M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, (1977), ii+149 pp.
[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.
[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), 527-530.
[22]	V. Kaloshin, K. Zhang and Y. Zheng, "Almost Dense Orbit on Energy Surface," Proceedings of the XVI-th ICMP, Prague, (2009), 314-322.
[23]	P. Lochak and J.-P. Marco, Diffusion times and stability exponents for nearly integrable analytic systems, Central European Journal of Mathematics, 3 (2005), 342-397. 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), viii+145 pp.
[25]	P. Lochak, Arnold diffusion; a compendium of remarks and questions, in "Hamiltonian Systems with Three or More Degrees of Freedom" (S'Agaró, 1995), 168-183, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, 1999.
[26]	J.-P. Marco, Uniform lower bounds of the splitting for analytic symplectic systems, preprint, 2005.
[27]	J.-P. Marco, Models for skew-products and polysystems, C. R. Acad. Sci. Paris, 346, (2008), 203-208.
[28]	J.-P. Marco, Arnold diffusion in a priori stable systems on $\A^3$, in preparation, 2010.
[29]	J.-P. Marco, Generic properties of classical systems on the torus $\T^2$, in preparation, 2010.
[30]	J.-P. Marco, Nets of hyperbolic annuli in generic nearly integrable systems on $\A^3$, in preparation, 2010.
[31]	J.-P. Marco, Skew-products and polysystems in the neighborhood of hyperbolic annuli, in preparation, 2010.
[32]	N. D. Mèzer, Arnol'd diffusion. I. Announcement of results, J. Math. Sci. (N. Y.), 124 (2004), 5275-5289. doi: 10.1023/B:JOTH.0000047353.78307.09.
[33]	R. Moeckel, Generic drift on Cantor sets of annuli, in "Celestial Mechanics" (Evanston, IL, 1999), 163-171, Contemp. Math., 292, Amer. Math. Soc., Providence, RI, 2002.
[34]	J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., II (1962), 1-20.
[35]	J. Moser, "Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics," Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, (1973), viii+198 pp.
[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), 199-275.
[37]	J.-P. Marco and D. Sauzin, Wandering domains and random walks in Gevrey near-integrable systems, Ergodic Theory Dynam. Systems, 24 (2004), 1619-1666. 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), 5–-66.
[39]	N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II, Trudy Sem. Petrovsk., 5 (1979), 5-50.
[40]	D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity, 17 (2004), 1803-1841. doi: 10.1088/0951-7715/17/5/014.
[41]	J.-C. Yoccoz, Introduction to hyperbolic dynamics, in "Real and Complex Dynamical Systems" (Hillerød, 1993), 265-291, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995.
[42]	K. Zhang, Speed of Arnold diffusion for analytic Hamiltonian systems, to appear in Inventiones Mathematicae, 2011.

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), 9-36. doi: 10.1070/RM1963v018n05ABEH004130.
[2]	V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12.
[3]	V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," [Dynamical Systems III], Transl. from the Russian original by E. Khukhro, Third edition, Encyclopedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 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), 613-664. 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), 189-195.
[6]	P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615–-669. 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), 1115-1135. 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), 97-112. 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.
[10]	A. Bounemoura, Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians, Journal of Differential Equations, 249 (2010), 2905-2920. doi: 10.1016/j.jde.2010.06.004.
[11]	L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. Henri Poincaré, Phys. Théor., 60 (1994), 144 pp.
[12]	J. Cresson and C. Guillet, Periodic orbits and Arnold diffusion, Discrete Contin. Dyn. Syst., 9 (2003), 451-470.
[13]	M. Chaperon, Stable manifolds and the Perron-Irwin method, Ergodic Theory Dynam. Systems, 24 (2004), 1359-1394. 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), 457-517.
[15]	C.-Q. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, Journal of Differential Geometry, 82 (2009), 229-277.
[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), viii+141 pp.
[17]	M. R. Herman, Some open problems in dynamical systems, in "Proceedings of the International Congress of Mathematicians," Vol. II (Berlin, 1998), Doc. Math., 1998, Extra Vol. II, 797-808.
[18]	M. Chaperon, The Lipschitzian core of some invariant manifold theorems, Ergodic Theory Dynam. Systems, 28 (2008), 1419-1441. doi: 10.1017/S0143385707000910.
[19]	M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, (1977), ii+149 pp.
[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.
[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), 527-530.
[22]	V. Kaloshin, K. Zhang and Y. Zheng, "Almost Dense Orbit on Energy Surface," Proceedings of the XVI-th ICMP, Prague, (2009), 314-322.
[23]	P. Lochak and J.-P. Marco, Diffusion times and stability exponents for nearly integrable analytic systems, Central European Journal of Mathematics, 3 (2005), 342-397. 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), viii+145 pp.
[25]	P. Lochak, Arnold diffusion; a compendium of remarks and questions, in "Hamiltonian Systems with Three or More Degrees of Freedom" (S'Agaró, 1995), 168-183, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, 1999.
[26]	J.-P. Marco, Uniform lower bounds of the splitting for analytic symplectic systems, preprint, 2005.
[27]	J.-P. Marco, Models for skew-products and polysystems, C. R. Acad. Sci. Paris, 346, (2008), 203-208.
[28]	J.-P. Marco, Arnold diffusion in a priori stable systems on $\A^3$, in preparation, 2010.
[29]	J.-P. Marco, Generic properties of classical systems on the torus $\T^2$, in preparation, 2010.
[30]	J.-P. Marco, Nets of hyperbolic annuli in generic nearly integrable systems on $\A^3$, in preparation, 2010.
[31]	J.-P. Marco, Skew-products and polysystems in the neighborhood of hyperbolic annuli, in preparation, 2010.
[32]	N. D. Mèzer, Arnol'd diffusion. I. Announcement of results, J. Math. Sci. (N. Y.), 124 (2004), 5275-5289. doi: 10.1023/B:JOTH.0000047353.78307.09.
[33]	R. Moeckel, Generic drift on Cantor sets of annuli, in "Celestial Mechanics" (Evanston, IL, 1999), 163-171, Contemp. Math., 292, Amer. Math. Soc., Providence, RI, 2002.
[34]	J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., II (1962), 1-20.
[35]	J. Moser, "Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics," Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, (1973), viii+198 pp.
[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), 199-275.
[37]	J.-P. Marco and D. Sauzin, Wandering domains and random walks in Gevrey near-integrable systems, Ergodic Theory Dynam. Systems, 24 (2004), 1619-1666. 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), 5–-66.
[39]	N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II, Trudy Sem. Petrovsk., 5 (1979), 5-50.
[40]	D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity, 17 (2004), 1803-1841. doi: 10.1088/0951-7715/17/5/014.
[41]	J.-C. Yoccoz, Introduction to hyperbolic dynamics, in "Real and Complex Dynamical Systems" (Hillerød, 1993), 265-291, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995.
[42]	K. Zhang, Speed of Arnold diffusion for analytic Hamiltonian systems, to appear in Inventiones Mathematicae, 2011.

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