-
Previous Article
Uniform inertial sets for damped wave equations
- DCDS Home
- This Issue
-
Next Article
Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations
Transition tori near an elliptic-fixed point
1. | Departmento de Matematicas, UAM-Iztapalapa, C. P. 09340, A. P. 55-534 Mexico DF, Mexico |
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.
[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] |
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 |
[6] |
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 |
[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] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
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 |
[17] |
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 |
[18] |
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 |
[19] |
Shin-Ichiro Ei, Hiroshi Matsuzawa. The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 901-921. doi: 10.3934/dcds.2010.26.901 |
[20] |
Graham W. Alldredge, Ruo Li, Weiming Li. Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry. Kinetic & Related Models, 2016, 9 (2) : 237-249. doi: 10.3934/krm.2016.9.237 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]