# American Institute of Mathematical Sciences

2012, 6(4): 499-538. doi: 10.3934/jmd.2012.6.499

## 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

Received  May 2012 Published  January 2013

We investigate the existence of Arnold diffusion-type orbits for systems obtained by iterating in any order the time-one maps of a family of Tonelli Hamiltonians. Such systems are known as 'polysystems' or 'iterated function systems'. When specialized to families of twist maps on the cylinder, our results are similar to those obtained by Moeckel [20] and Le Calvez [15]. Our approach is based on weak KAM theory and is close to the one used by Bernard in [3] to study the case of a single Tonelli Hamiltonian.
Citation: Vito Mandorino. Connecting orbits for families of Tonelli Hamiltonians. Journal of Modern Dynamics, 2012, 6 (4) : 499-538. doi: 10.3934/jmd.2012.6.499
##### 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, (). [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). [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., (). [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.

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##### 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, (). [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). [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., (). [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.
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