October  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), 9-12.

[2]

Patrick Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier (Grenoble), 52 (2002), 1533-1568.

[3]

_____, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615-669. 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), 369-386.

[5]

_____, Sur quelques courbes fermées remarquables, Bull. Soc. Math. France, 60 (1932), 1-26.

[6]

Abed Bounemoura and Edouard Pennamen, Instability for a priori unstable Hamiltonian systems: A dynamical approach, Discrete Contin. Dyn. Syst., 32 (2012), 753-793. 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, Birkhäuser Boston, Inc., Boston, MA, 2004.

[8]

Chong-Qing Cheng and Jun Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom., 67 (2004), 457-517.

[9]

_____, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277.

[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), 4104-4123. doi: 10.1016/j.jde.2011.01.020.

[12]

Albert Fathi, Weak KAM Theorem and Lagrangian dynamics, Preliminary version number 10 - version 15, June, 2008.

[13]

Boris Hasselblatt and Anatole Katok, Principal structures, in "Handbook of Dynamical Systems," Vol. 1A, North-Holland, Amsterdam, (2002), 1-203. 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), 869-879. doi: 10.1017/S0143385706000903.

[16]

Jean-Pierre Marco, Modèles pour les applications fibrées et les polysystèmes, C. R. Math. Acad. Sci. Paris, 346 (2008), 203-208. 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), 169-207. doi: 10.1007/BF02571383.

[18]

_____, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., 4 (1991), 207-263. doi: 10.2307/2939275.

[19]

_____, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.

[20]

Richard Moeckel, Generic drift on Cantor sets of annuli, in "Celestial Mechanics" (Evanston, IL, 1999), Contemp. Math., 292, Amer. Math. Soc., Providence, RI, (2002), 163-171. doi: 10.1090/conm/292/04922.

[21]

Jürgen Moser, Monotone twist mappings and the calculus of variations, Ergodic Theory Dynam. Systems, 6 (1986), 401-413. doi: 10.1017/S0143385700003588.

[22]

Maxime Zavidovique, Weak KAM for commuting Hamiltonians, Nonlinearity, 23 (2010), 793-808. 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), 1-39.

show all references

References:
[1]

V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12.

[2]

Patrick Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier (Grenoble), 52 (2002), 1533-1568.

[3]

_____, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615-669. 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), 369-386.

[5]

_____, Sur quelques courbes fermées remarquables, Bull. Soc. Math. France, 60 (1932), 1-26.

[6]

Abed Bounemoura and Edouard Pennamen, Instability for a priori unstable Hamiltonian systems: A dynamical approach, Discrete Contin. Dyn. Syst., 32 (2012), 753-793. 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, Birkhäuser Boston, Inc., Boston, MA, 2004.

[8]

Chong-Qing Cheng and Jun Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom., 67 (2004), 457-517.

[9]

_____, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277.

[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), 4104-4123. doi: 10.1016/j.jde.2011.01.020.

[12]

Albert Fathi, Weak KAM Theorem and Lagrangian dynamics, Preliminary version number 10 - version 15, June, 2008.

[13]

Boris Hasselblatt and Anatole Katok, Principal structures, in "Handbook of Dynamical Systems," Vol. 1A, North-Holland, Amsterdam, (2002), 1-203. 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), 869-879. doi: 10.1017/S0143385706000903.

[16]

Jean-Pierre Marco, Modèles pour les applications fibrées et les polysystèmes, C. R. Math. Acad. Sci. Paris, 346 (2008), 203-208. 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), 169-207. doi: 10.1007/BF02571383.

[18]

_____, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., 4 (1991), 207-263. doi: 10.2307/2939275.

[19]

_____, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.

[20]

Richard Moeckel, Generic drift on Cantor sets of annuli, in "Celestial Mechanics" (Evanston, IL, 1999), Contemp. Math., 292, Amer. Math. Soc., Providence, RI, (2002), 163-171. doi: 10.1090/conm/292/04922.

[21]

Jürgen Moser, Monotone twist mappings and the calculus of variations, Ergodic Theory Dynam. Systems, 6 (1986), 401-413. doi: 10.1017/S0143385700003588.

[22]

Maxime Zavidovique, Weak KAM for commuting Hamiltonians, Nonlinearity, 23 (2010), 793-808. 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), 1-39.

[1]

Renato Iturriaga, Héctor Sánchez Morgado. The Lax-Oleinik semigroup on graphs. Networks and Heterogeneous Media, 2017, 12 (4) : 643-662. doi: 10.3934/nhm.2017026

[2]

Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61

[3]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57

[4]

Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545

[5]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069

[6]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080

[7]

Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693

[8]

Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451

[9]

Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475

[10]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114

[11]

Saisai Shi, Bo Tan, Qinglong Zhou. Best approximation of orbits in iterated function systems. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4085-4104. doi: 10.3934/dcds.2021029

[12]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097

[13]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106

[14]

Jacky Cresson. The transfer lemma for Graff tori and Arnold diffusion time. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 787-800. doi: 10.3934/dcds.2001.7.787

[15]

Massimiliano Berti, Philippe Bolle. Fast Arnold diffusion in systems with three time scales. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 795-811. doi: 10.3934/dcds.2002.8.795

[16]

Massimiliano Berti. Some remarks on a variational approach to Arnold's diffusion. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 307-314. doi: 10.3934/dcds.1996.2.307

[17]

Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555

[18]

Peng Chen, Linfeng Mei, Xianhua Tang. Nonstationary homoclinic orbit for an infinite-dimensional fractional reaction-diffusion system. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021279

[19]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235

[20]

Pablo G. Barrientos, Abbas Fakhari, Aliasghar Sarizadeh. Density of fiberwise orbits in minimal iterated function systems on the circle. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3341-3352. doi: 10.3934/dcds.2014.34.3341

2020 Impact Factor: 0.848

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]