American Institute of Mathematical Sciences

Advanced
April  2000, 6(2): 381-392. doi: 10.3934/dcds.2000.6.381

Transition tori near an elliptic-fixed point

Antonio Garcia 1,

1. 

Departmento de Matematicas, UAM-Iztapalapa, C. P. 09340, A. P. 55-534 Mexico DF, Mexico

Revised  August 1999 Published  January 2000

Let $F:(M,\omega) \mapsto (M,\omega)$ be a smooth symplectic diffeomorphism with a fixed point a and a heteroclinic orbit in the sense of been in the intersection of the central stable and the central unstable manifolds of the fixed point. It is studied the case when the tangent space of a point in the heteroclinic orbit is the direct sum of three subspaces. The first one is the characteristic bundle of the central stable manifold of $\mathbf a$ the second one is the characteristic bundle of the central unstable manifold of $\mathbf a$, and the third one is tangent to the intersection of the central stable and unstable manifolds.
In this situation, the homoclinic map $\Lambda$ is a smooth and symplectic diffeomorphism of open subsets of the central manifold of $\mathbf a$.
Moreover, if an invariant circle intersects the domain of definition of $\Lambda$ and its image intersects other circle, there are orbits that wander from one circle to the other. This phenomenon is similar to the Arnold diffusion.
The Melnikov Method gives sufficient conditions for the existence of homoclinic maps, and non identity homoclinic maps in a perturbation of a Hamiltonian system.
Keywords: Arnold Diffusion, Symplectic geometry, transition tori, Melnikov method..
Mathematics Subject Classification: 58F05,70H0.
Citation: Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381
[1]

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
[2]

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
[3]

Matteo Novaga, Enrico Valdinoci. The geometry of mesoscopic phase transition interfaces. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 777-798. doi: 10.3934/dcds.2007.19.777
[4]

Fiammetta Battaglia and Elisa Prato. Nonrational, nonsimple convex polytopes in symplectic geometry. Electronic Research Announcements, 2002, 8: 29-34.

[5]

Marian Gidea, Clark Robinson. Obstruction argument for transition chains of tori interspersed with gaps. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 393-416. doi: 10.3934/dcdss.2009.2.393
[6]

Kazuyuki Yagasaki. Application of the subharmonic Melnikov method to piecewise-smooth systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2189-2209. doi: 10.3934/dcds.2013.33.2189
[7]

I-Liang Chern, Chun-Hsiung Hsia. Dynamic phase transition for binary systems in cylindrical geometry. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 173-188. doi: 10.3934/dcdsb.2011.16.173
[8]

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
[9]

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
[10]

Massimiliano Berti. Some remarks on a variational approach to Arnold's diffusion. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 307-314. doi: 10.3934/dcds.1996.2.307
[11]

Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431
[12]

Carlos Durán, Diego Otero. The projective symplectic geometry of higher order variational problems: Minimality conditions. Journal of Geometric Mechanics, 2016, 8 (3) : 305-322. doi: 10.3934/jgm.2016009
[13]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020196
[14]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020184
[15]

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
[16]

Amadeu Delshams, Rodrigo G. Schaefer. Arnold diffusion for a complete family of perturbations with two independent harmonics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6047-6072. doi: 10.3934/dcds.2018261
[17]

Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[18]

Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105
[19]

Michio Urano, Kimie Nakashima, Yoshio Yamada. Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity. Conference Publications, 2005, 2005 (Special) : 868-877. doi: 10.3934/proc.2005.2005.868
[20]

Alethea B. T. Barbaro, Pierre Degond. Phase transition and diffusion among socially interacting self-propelled agents. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1249-1278. doi: 10.3934/dcdsb.2014.19.1249

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences