Transition map and shadowing lemma for normally hyperbolic invariant manifolds

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March 2013, 33(3): 1089-1112. doi: 10.3934/dcds.2013.33.1089

Transition map and shadowing lemma for normally hyperbolic invariant manifolds

Amadeu Delshams ^1, , Marian Gidea ^2, and Pablo Roldán ^3,

1.	Departament de Matemática Aplicada I, ETSEIB-UPC, 08028 Barcelona
2.	School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, United States
3.	Departament de Matemática Aplicada I, ETSEIB-UPC, 08028 Barcelona,, Spain

Received April 2011 Revised December 2011 Published October 2012

Abstract
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For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in a Hamiltonian system. This consists in the following steps: (i) computation of the scattering map and of the transition map for the Hamiltonian flow, (ii) reduction to the scattering map and to the transition map, respectively, for the return map to some surface of section, (iii) construction of sequences of windows within the surface of section, with the successive pairs of windows correctly aligned, alternately, under the transition map, and under some power of the inner map, (iv) detection of trajectories which follow closely those windows. We illustrate this strategy with two models: the large gap problem for nearly integrable Hamiltonian systems, and the the spatial circular restricted three-body problem.

Keywords: Hamiltonian instability, the three-body problem., Arnold diffusion.

Mathematics Subject Classification: Primary: 37J40, 37C50, 37C29; Secondary: 37B3.

Citation: Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089

References:

[1]	V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady, 5 (1964), 581-585.
[2]	K. Burns and M. Gidea, "Differential Geometry and Topology. With a View to Dynamical Systems,'' Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 2005.
[3]	J. Cresson and C. Guillet, Hyperbolicity versus partial-hyperbolicity and the transversality-torsion phenomenon, J. Differential Equations, 244 (2008), 2123-2132. doi: 10.1016/j.jde.2008.02.009.
[4]	A. Delshams, M. Gidea and P. Roldan, Arnold's mechanism of diffusion in the spatial circular restricted three-body problem: A semi-numerical argument, preprint, 2010.
[5]	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 $T^2$, Comm. Math. Phys., 209 (2000), 353-392.
[6]	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, Mem. Amer. Math. Soc., 179 (2006), viii+141 pp.
[7]	A. Delshams, R. de la Llave and T. M. Seara, Geometric properties of the scattering map of a normally hyperbolic invariant manifold, Adv. Math., 217 (2008), 1096-1153. doi: 10.1016/j.aim.2007.08.014.
[8]	A. Delshams, M. Gidea, R. de la Llave and T. M. Seara, Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation, in (Hamiltonian dynamical systems and applications), NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, (2008), 285-336.
[9]	A. Delshams, J. Masdemont and P. Roldán, Computing the scattering map in the spatial Hill's problem, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 455-483. doi: 10.3934/dcdsb.2008.10.455.
[10]	R. W. Easton, Homoclinic phenomena in Hamiltonian systems with several degrees of freedom, J. Differential Equations, 29 (1978), 241-252. doi: 10.1016/0022-0396(78)90123-7.
[11]	N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1973/74), 1109-1137.
[12]	A. García, Transition tori near an elliptic fixed point, Discrete Contin. Dynam. Systems, 6 (2000), 381-392.
[13]	M. Gidea and R. de la Llave, Topological methods in the instability problem of Hamiltonian systems, Discrete Contin. Dyn. Syst., 14 (2006), 295-328.
[14]	M. Gidea and C. Robinson, Topologically crossing heteroclinic connections to invariant tori, J. Differential Equations, 193 (2003), 49-74. doi: 10.1016/S0022-0396(03)00065-2.
[15]	M. Gidea and C. Robinson, Obstruction argument for transition chains of tori interspersed with gaps, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 393-416.
[16]	M. Gidea, and C. Robinson, Diffusion along transition chains of invariant tori and Aubry-Mather sets, Ergodic Theory and Dynamical Systems, DOI: 10.1017/S0143385712000363. doi: 10.1017/S0143385712000363.
[17]	M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin, 1977.
[18]	H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), 148 (1998), 197-289.
[19]	J. P. Marco, A normally hyperbolic lambda lemma with applications to diffusion, Preprint, 2008.
[20]	C. Pugh and M. Shub, Linearization of normally hyperbolic diffeomorphisms and flows, Invent. Math., 10 (1970), 187-198. doi: 10.1007/BF01403247.
[21]	C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,'' Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, second edition, 1999.
[22]	P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58. doi: 10.1016/j.jde.2004.03.013.

show all references

References:

[1]	V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady, 5 (1964), 581-585.
[2]	K. Burns and M. Gidea, "Differential Geometry and Topology. With a View to Dynamical Systems,'' Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 2005.
[3]	J. Cresson and C. Guillet, Hyperbolicity versus partial-hyperbolicity and the transversality-torsion phenomenon, J. Differential Equations, 244 (2008), 2123-2132. doi: 10.1016/j.jde.2008.02.009.
[4]	A. Delshams, M. Gidea and P. Roldan, Arnold's mechanism of diffusion in the spatial circular restricted three-body problem: A semi-numerical argument, preprint, 2010.
[5]	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 $T^2$, Comm. Math. Phys., 209 (2000), 353-392.
[6]	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, Mem. Amer. Math. Soc., 179 (2006), viii+141 pp.
[7]	A. Delshams, R. de la Llave and T. M. Seara, Geometric properties of the scattering map of a normally hyperbolic invariant manifold, Adv. Math., 217 (2008), 1096-1153. doi: 10.1016/j.aim.2007.08.014.
[8]	A. Delshams, M. Gidea, R. de la Llave and T. M. Seara, Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation, in (Hamiltonian dynamical systems and applications), NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, (2008), 285-336.
[9]	A. Delshams, J. Masdemont and P. Roldán, Computing the scattering map in the spatial Hill's problem, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 455-483. doi: 10.3934/dcdsb.2008.10.455.
[10]	R. W. Easton, Homoclinic phenomena in Hamiltonian systems with several degrees of freedom, J. Differential Equations, 29 (1978), 241-252. doi: 10.1016/0022-0396(78)90123-7.
[11]	N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1973/74), 1109-1137.
[12]	A. García, Transition tori near an elliptic fixed point, Discrete Contin. Dynam. Systems, 6 (2000), 381-392.
[13]	M. Gidea and R. de la Llave, Topological methods in the instability problem of Hamiltonian systems, Discrete Contin. Dyn. Syst., 14 (2006), 295-328.
[14]	M. Gidea and C. Robinson, Topologically crossing heteroclinic connections to invariant tori, J. Differential Equations, 193 (2003), 49-74. doi: 10.1016/S0022-0396(03)00065-2.
[15]	M. Gidea and C. Robinson, Obstruction argument for transition chains of tori interspersed with gaps, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 393-416.
[16]	M. Gidea, and C. Robinson, Diffusion along transition chains of invariant tori and Aubry-Mather sets, Ergodic Theory and Dynamical Systems, DOI: 10.1017/S0143385712000363. doi: 10.1017/S0143385712000363.
[17]	M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin, 1977.
[18]	H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), 148 (1998), 197-289.
[19]	J. P. Marco, A normally hyperbolic lambda lemma with applications to diffusion, Preprint, 2008.
[20]	C. Pugh and M. Shub, Linearization of normally hyperbolic diffeomorphisms and flows, Invent. Math., 10 (1970), 187-198. doi: 10.1007/BF01403247.
[21]	C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,'' Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, second edition, 1999.
[22]	P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58. doi: 10.1016/j.jde.2004.03.013.

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