# American Institute of Mathematical Sciences

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

## Transition map and shadowing lemma for normally hyperbolic invariant manifolds

 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

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.
Citation: Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & 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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [11] N. Fenichel, Asymptotic stability with rate conditions,, Indiana Univ. Math. J., 23 (): 1109.   Google Scholar [12] A. García, Transition tori near an elliptic fixed point, Discrete Contin. Dynam. Systems, 6 (2000), 381-392.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin, 1977.  Google Scholar [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.  Google Scholar [19] J. P. Marco, A normally hyperbolic lambda lemma with applications to diffusion, Preprint, 2008. Google Scholar [20] C. Pugh and M. Shub, Linearization of normally hyperbolic diffeomorphisms and flows, Invent. Math., 10 (1970), 187-198. doi: 10.1007/BF01403247.  Google Scholar [21] C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,'' Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, second edition, 1999.  Google Scholar [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.  Google Scholar

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##### References:
 [1] V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady, 5 (1964), 581-585. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [11] N. Fenichel, Asymptotic stability with rate conditions,, Indiana Univ. Math. J., 23 (): 1109.   Google Scholar [12] A. García, Transition tori near an elliptic fixed point, Discrete Contin. Dynam. Systems, 6 (2000), 381-392.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin, 1977.  Google Scholar [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.  Google Scholar [19] J. P. Marco, A normally hyperbolic lambda lemma with applications to diffusion, Preprint, 2008. Google Scholar [20] C. Pugh and M. Shub, Linearization of normally hyperbolic diffeomorphisms and flows, Invent. Math., 10 (1970), 187-198. doi: 10.1007/BF01403247.  Google Scholar [21] C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,'' Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, second edition, 1999.  Google Scholar [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.  Google Scholar
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