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Connecting orbits for families of Tonelli Hamiltonians
1. | Université Paris-Dauphine, CEREMADE UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16 |
References:
[1] |
V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom,, Dokl. Akad. Nauk SSSR, 156 (1964), 9.
|
[2] |
Patrick Bernard, Connecting orbits of time dependent Lagrangian systems,, Ann. Inst. Fourier (Grenoble), 52 (2002), 1533.
|
[3] |
_____, The dynamics of pseudographs in convex Hamiltonian systems,, J. Amer. Math. Soc., 21 (2008), 615.
doi: 10.1090/S0894-0347-08-00591-2. |
[4] |
George D. Birkhoff, Sur l'existence de régions d'instabilité en Dynamique,, Ann. Inst. H. Poincaré, 2 (1932), 369.
|
[5] |
_____, Sur quelques courbes fermées remarquables,, Bull. Soc. Math. France, 60 (1932), 1.
|
[6] |
Abed Bounemoura and Edouard Pennamen, Instability for a priori unstable Hamiltonian systems: A dynamical approach,, Discrete Contin. Dyn. Syst., 32 (2012), 753.
doi: 10.3934/dcds.2012.32.753. |
[7] |
Piermarco Cannarsa and Carlo Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,", Progress in Nonlinear Differential Equations and their Applications, 58 (2004).
|
[8] |
Chong-Qing Cheng and Jun Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems,, J. Differential Geom., 67 (2004), 457.
|
[9] |
_____, Arnold diffusion in Hamiltonian systems: A priori unstable case,, J. Differential Geom., 82 (2009), 229.
|
[10] |
X. Cui, On commuting Tonelli Hamiltonians: Time-periodic case,, preprint, (). Google Scholar |
[11] |
Xiaojun Cui and Ji Li, On commuting Tonelli Hamiltonians: Autonomous case,, Journal of Differential Equations, 250 (2011), 4104.
doi: 10.1016/j.jde.2011.01.020. |
[12] |
Albert Fathi, Weak KAM Theorem and Lagrangian dynamics,, Preliminary version number 10 - version 15, (2008). Google Scholar |
[13] |
Boris Hasselblatt and Anatole Katok, Principal structures,, in, (2002), 1.
doi: 10.1016/S1874-575X(02)80003-0. |
[14] |
, Olivier Jaulent,, Ph.D. thesis., (). Google Scholar |
[15] |
Patrice Le Calvez, Drift orbits for families of twist maps of the annulus,, Ergodic Theory Dynam. Systems, 27 (2007), 869.
doi: 10.1017/S0143385706000903. |
[16] |
Jean-Pierre Marco, Modèles pour les applications fibrées et les polysystèmes,, C. R. Math. Acad. Sci. Paris, 346 (2008), 203.
doi: 10.1016/j.crma.2007.11.017. |
[17] |
John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems,, Math. Z., 207 (1991), 169.
doi: 10.1007/BF02571383. |
[18] |
_____, Variational construction of orbits of twist diffeomorphisms,, J. Amer. Math. Soc., 4 (1991), 207.
doi: 10.2307/2939275. |
[19] |
_____, Variational construction of connecting orbits,, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349.
|
[20] |
Richard Moeckel, Generic drift on Cantor sets of annuli,, in, 292 (2002), 163.
doi: 10.1090/conm/292/04922. |
[21] |
Jürgen Moser, Monotone twist mappings and the calculus of variations,, Ergodic Theory Dynam. Systems, 6 (1986), 401.
doi: 10.1017/S0143385700003588. |
[22] |
Maxime Zavidovique, Weak KAM for commuting Hamiltonians,, Nonlinearity, 23 (2010), 793.
doi: 10.1088/0951-7715/23/4/002. |
[23] |
_____, Strict sub-solutions and Mañé potential in discrete weak KAM theory,, Commentarii Mathematici Elvetici, 87 (2012), 1. Google Scholar |
show all references
References:
[1] |
V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom,, Dokl. Akad. Nauk SSSR, 156 (1964), 9.
|
[2] |
Patrick Bernard, Connecting orbits of time dependent Lagrangian systems,, Ann. Inst. Fourier (Grenoble), 52 (2002), 1533.
|
[3] |
_____, The dynamics of pseudographs in convex Hamiltonian systems,, J. Amer. Math. Soc., 21 (2008), 615.
doi: 10.1090/S0894-0347-08-00591-2. |
[4] |
George D. Birkhoff, Sur l'existence de régions d'instabilité en Dynamique,, Ann. Inst. H. Poincaré, 2 (1932), 369.
|
[5] |
_____, Sur quelques courbes fermées remarquables,, Bull. Soc. Math. France, 60 (1932), 1.
|
[6] |
Abed Bounemoura and Edouard Pennamen, Instability for a priori unstable Hamiltonian systems: A dynamical approach,, Discrete Contin. Dyn. Syst., 32 (2012), 753.
doi: 10.3934/dcds.2012.32.753. |
[7] |
Piermarco Cannarsa and Carlo Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,", Progress in Nonlinear Differential Equations and their Applications, 58 (2004).
|
[8] |
Chong-Qing Cheng and Jun Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems,, J. Differential Geom., 67 (2004), 457.
|
[9] |
_____, Arnold diffusion in Hamiltonian systems: A priori unstable case,, J. Differential Geom., 82 (2009), 229.
|
[10] |
X. Cui, On commuting Tonelli Hamiltonians: Time-periodic case,, preprint, (). Google Scholar |
[11] |
Xiaojun Cui and Ji Li, On commuting Tonelli Hamiltonians: Autonomous case,, Journal of Differential Equations, 250 (2011), 4104.
doi: 10.1016/j.jde.2011.01.020. |
[12] |
Albert Fathi, Weak KAM Theorem and Lagrangian dynamics,, Preliminary version number 10 - version 15, (2008). Google Scholar |
[13] |
Boris Hasselblatt and Anatole Katok, Principal structures,, in, (2002), 1.
doi: 10.1016/S1874-575X(02)80003-0. |
[14] |
, Olivier Jaulent,, Ph.D. thesis., (). Google Scholar |
[15] |
Patrice Le Calvez, Drift orbits for families of twist maps of the annulus,, Ergodic Theory Dynam. Systems, 27 (2007), 869.
doi: 10.1017/S0143385706000903. |
[16] |
Jean-Pierre Marco, Modèles pour les applications fibrées et les polysystèmes,, C. R. Math. Acad. Sci. Paris, 346 (2008), 203.
doi: 10.1016/j.crma.2007.11.017. |
[17] |
John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems,, Math. Z., 207 (1991), 169.
doi: 10.1007/BF02571383. |
[18] |
_____, Variational construction of orbits of twist diffeomorphisms,, J. Amer. Math. Soc., 4 (1991), 207.
doi: 10.2307/2939275. |
[19] |
_____, Variational construction of connecting orbits,, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349.
|
[20] |
Richard Moeckel, Generic drift on Cantor sets of annuli,, in, 292 (2002), 163.
doi: 10.1090/conm/292/04922. |
[21] |
Jürgen Moser, Monotone twist mappings and the calculus of variations,, Ergodic Theory Dynam. Systems, 6 (1986), 401.
doi: 10.1017/S0143385700003588. |
[22] |
Maxime Zavidovique, Weak KAM for commuting Hamiltonians,, Nonlinearity, 23 (2010), 793.
doi: 10.1088/0951-7715/23/4/002. |
[23] |
_____, Strict sub-solutions and Mañé potential in discrete weak KAM theory,, Commentarii Mathematici Elvetici, 87 (2012), 1. Google Scholar |
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