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

Connecting orbits for families of Tonelli Hamiltonians

Vito Mandorino 1,

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.
Keywords: heteroclinic orbit, Lax--Oleinik operator., Arnold diffusion, iterated function system, weak KAM theory, Tonelli Hamiltonian.
Mathematics Subject Classification: Primary: 37J50, 37J45; Secondary: 37J40, 70Hx.
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.   Google Scholar

[2]

Patrick Bernard, Connecting orbits of time dependent Lagrangian systems,, Ann. Inst. Fourier (Grenoble), 52 (2002), 1533.   Google Scholar

[3]

_____, The dynamics of pseudographs in convex Hamiltonian systems,, J. Amer. Math. Soc., 21 (2008), 615.  doi: 10.1090/S0894-0347-08-00591-2.  Google Scholar

[4]

George D. Birkhoff, Sur l'existence de régions d'instabilité en Dynamique,, Ann. Inst. H. Poincaré, 2 (1932), 369.   Google Scholar

[5]

_____, Sur quelques courbes fermées remarquables,, Bull. Soc. Math. France, 60 (1932), 1.   Google Scholar

[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.  Google Scholar

[7]

Piermarco Cannarsa and Carlo Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,", Progress in Nonlinear Differential Equations and their Applications, 58 (2004).   Google Scholar

[8]

Chong-Qing Cheng and Jun Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems,, J. Differential Geom., 67 (2004), 457.   Google Scholar

[9]

_____, Arnold diffusion in Hamiltonian systems: A priori unstable case,, J. Differential Geom., 82 (2009), 229.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems,, Math. Z., 207 (1991), 169.  doi: 10.1007/BF02571383.  Google Scholar

[18]

_____, Variational construction of orbits of twist diffeomorphisms,, J. Amer. Math. Soc., 4 (1991), 207.  doi: 10.2307/2939275.  Google Scholar

[19]

_____, Variational construction of connecting orbits,, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349.   Google Scholar

[20]

Richard Moeckel, Generic drift on Cantor sets of annuli,, in, 292 (2002), 163.  doi: 10.1090/conm/292/04922.  Google Scholar

[21]

Jürgen Moser, Monotone twist mappings and the calculus of variations,, Ergodic Theory Dynam. Systems, 6 (1986), 401.  doi: 10.1017/S0143385700003588.  Google Scholar

[22]

Maxime Zavidovique, Weak KAM for commuting Hamiltonians,, Nonlinearity, 23 (2010), 793.  doi: 10.1088/0951-7715/23/4/002.  Google Scholar

[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.   Google Scholar

[2]

Patrick Bernard, Connecting orbits of time dependent Lagrangian systems,, Ann. Inst. Fourier (Grenoble), 52 (2002), 1533.   Google Scholar

[3]

_____, The dynamics of pseudographs in convex Hamiltonian systems,, J. Amer. Math. Soc., 21 (2008), 615.  doi: 10.1090/S0894-0347-08-00591-2.  Google Scholar

[4]

George D. Birkhoff, Sur l'existence de régions d'instabilité en Dynamique,, Ann. Inst. H. Poincaré, 2 (1932), 369.   Google Scholar

[5]

_____, Sur quelques courbes fermées remarquables,, Bull. Soc. Math. France, 60 (1932), 1.   Google Scholar

[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.  Google Scholar

[7]

Piermarco Cannarsa and Carlo Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,", Progress in Nonlinear Differential Equations and their Applications, 58 (2004).   Google Scholar

[8]

Chong-Qing Cheng and Jun Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems,, J. Differential Geom., 67 (2004), 457.   Google Scholar

[9]

_____, Arnold diffusion in Hamiltonian systems: A priori unstable case,, J. Differential Geom., 82 (2009), 229.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems,, Math. Z., 207 (1991), 169.  doi: 10.1007/BF02571383.  Google Scholar

[18]

_____, Variational construction of orbits of twist diffeomorphisms,, J. Amer. Math. Soc., 4 (1991), 207.  doi: 10.2307/2939275.  Google Scholar

[19]

_____, Variational construction of connecting orbits,, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349.   Google Scholar

[20]

Richard Moeckel, Generic drift on Cantor sets of annuli,, in, 292 (2002), 163.  doi: 10.1090/conm/292/04922.  Google Scholar

[21]

Jürgen Moser, Monotone twist mappings and the calculus of variations,, Ergodic Theory Dynam. Systems, 6 (1986), 401.  doi: 10.1017/S0143385700003588.  Google Scholar

[22]

Maxime Zavidovique, Weak KAM for commuting Hamiltonians,, Nonlinearity, 23 (2010), 793.  doi: 10.1088/0951-7715/23/4/002.  Google Scholar

[23]

_____, Strict sub-solutions and Mañé potential in discrete weak KAM theory,, Commentarii Mathematici Elvetici, 87 (2012), 1.   Google Scholar

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