American Institute of Mathematical Sciences

Advanced
January  2001, 7(1): 61-84. doi: 10.3934/dcds.2001.7.61

Arnold diffusion in perturbations of analytic integrable Hamiltonian systems

Ernest Fontich 1, and  Pau Martín 2,

1. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain
2. 

Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Ed-C3, Jordi Girona, 1-3, 08034 Barcelona, Spain

Received  June 1999 Revised  October 2000 Published  November 2000

Given an analytic integrable Hamiltonian with three or more degrees of freedom, we construct, arbitrarily close to it, an analytic perturbation with transition chains whose lengths only depend on the unperturbed Hamiltonian. Then we deduce that the perturbed system has Arnold diffusion. We provide the technical details of the tools we use.
Keywords: Hamiltonian systems, heteroclinic solutions, invariant tori, Arnold diffusion..
Mathematics Subject Classification: 37J40, 37D10, 70H0.
Citation: Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61
[1]

Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371
[2]

C. Chandre. Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 457-465. doi: 10.3934/dcdsb.2002.2.457
[3]

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

Guowei Yu. Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4769-4793. doi: 10.3934/dcds.2013.33.4769
[5]

Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633
[6]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097
[7]

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

Denis G. Gaidashev. Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 63-102. doi: 10.3934/dcds.2005.13.63
[9]

Xiaocai Wang. Non-floquet invariant tori in reversible systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3439-3457. doi: 10.3934/dcds.2018147
[10]

Lorenzo Arona, Josep J. Masdemont. Computation of heteroclinic orbits between normally hyperbolic invariant 3-spheres foliated by 2-dimensional invariant Tori in Hill's problem. Conference Publications, 2007, 2007 (Special) : 64-74. doi: 10.3934/proc.2007.2007.64
[11]

Claude Froeschlé, Massimiliano Guzzo, Elena Lega. First numerical evidence of global Arnold diffusion in quasi-integrable systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 687-698. doi: 10.3934/dcdsb.2005.5.687
[12]

Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451
[13]

Shengqing Hu. Persistence of invariant tori for almost periodically forced reversible systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4497-4518. doi: 10.3934/dcds.2020188
[14]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
[15]

Francesca Alessio, Carlo Carminati, Piero Montecchiari. Heteroclinic motions joining almost periodic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 569-584. doi: 10.3934/dcds.1999.5.569
[16]

Amadeu Delshams, Pere Gutiérrez, Tere M. Seara. Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 785-826. doi: 10.3934/dcds.2004.11.785
[17]

Amadeu Delshams, Pere Gutiérrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 757-783. doi: 10.3934/dcds.2004.11.757
[18]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569
[19]

Xuemei Li, Zaijiu Shang. On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4225-4257. doi: 10.3934/dcds.2019171
[20]

Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences