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.   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

[1]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[2]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[3]

Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020054

[4]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[5]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015

[6]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[7]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003

[8]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127

[9]

Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019

[10]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021002

[11]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[12]

Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031

[13]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049

[14]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[15]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228

[16]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[17]

Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161

[18]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400

[19]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

[20]

Yuanshi Wang. Asymmetric diffusion in a two-patch mutualism system characterizing exchange of resource for resource. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 963-985. doi: 10.3934/dcdsb.2020149

2019 Impact Factor: 0.465

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]