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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 |
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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