July  2002, 8(3): 795-811. doi: 10.3934/dcds.2002.8.795

Fast Arnold diffusion in systems with three time scales

1. 

S.I.S.S.A., Via Beirut 2-4, 34014 Trieste, Italy

2. 

Département de mathématiques, Université d'Avignon, 33, rue Louis Pasteur, 84000 Avignon, France

Received  April 2001 Revised  August 2001 Published  April 2002

We consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the "splitting determinant" is exponentially small.
Citation: 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
[1]

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

[2]

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

[3]

Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039

[4]

Sergei Yu. Pilyugin. Variational shadowing. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 733-737. doi: 10.3934/dcdsb.2010.14.733

[5]

E. Canalias, Josep J. Masdemont. Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 261-279. doi: 10.3934/dcds.2006.14.261

[6]

Pau Martín, David Sauzin, Tere M. Seara. Exponentially small splitting of separatrices in the perturbed McMillan map. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 301-372. doi: 10.3934/dcds.2011.31.301

[7]

Karsten Matthies. Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 585-602. doi: 10.3934/dcds.2003.9.585

[8]

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

[9]

Sergey V. Bolotin. Shadowing chains of collision orbits. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 235-260. doi: 10.3934/dcds.2006.14.235

[10]

Jianquan Li, Yanni Xiao, Yali Yang. Global analysis of a simple parasite-host model with homoclinic orbits. Mathematical Biosciences & Engineering, 2012, 9 (4) : 767-784. doi: 10.3934/mbe.2012.9.767

[11]

W.R. Derrick, P. van den Driessche. Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 299-309. doi: 10.3934/dcdsb.2003.3.299

[12]

S. Yu. Pilyugin. Inverse shadowing by continuous methods. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 29-38. doi: 10.3934/dcds.2002.8.29

[13]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079

[14]

Amadeu Delshams, Vassili Gelfreich, Angel Jorba and Tere M. Seara. Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing. Electronic Research Announcements, 1997, 3: 1-10.

[15]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133

[16]

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

[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]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967

[19]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]