American Institute of Mathematical Sciences

Advanced
October  2004, 11(4): 785-826. doi: 10.3934/dcds.2004.11.785

Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds

Amadeu Delshams 1, , Pere Gutiérrez 2, and  Tere M. Seara 3,

1. 

Departament de Matemàtica Aplicada I, ETSEIB-Universitat Politècnica de Catalunya, Diagonal 647, E-08028 Barcelona
2. 

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028
3. 

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

Received  March 2003 Revised  January 2004 Published  September 2004

We consider a singular or weakly hyperbolic Hamiltonian, with $n+1$ degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The model consists of an integrable Hamiltonian possessing an $n$-dimensional hyperbolic invariant torus with fast frequencies $\omega/\sqrt\varepsilon$ and coincident whiskers, plus a perturbation of order $\mu=\varepsilon^p$. The vector $\omega$ is assumed to satisfy a Diophantine condition.
We provide a tool to study, in this singular case, the splitting of the perturbed whiskers for $\varepsilon$ small enough, as well as their homoclinic intersections, using the Poincaré--Melnikov method. Due to the exponential smallness of the Melnikov function, the size of the error term has to be carefully controlled. So we introduce flow-box coordinates in order to take advantage of the quasiperiodicity properties of the splitting. As a direct application of this approach, we obtain quite general upper bounds for the splitting.
Keywords: flow-box coordinates, Poincaré–Melnikov method, Hyperbolic KAM theory.
Mathematics Subject Classification: 70H08, 34C37, 34C30, 70K4.
Citation: Amadeu Delshams, Pere Gutiérrez, Tere M. Seara. Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 785-826. doi: 10.3934/dcds.2004.11.785
[1]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic and Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345
[2]

Wen Si, Fenfen Wang, Jianguo Si. Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method. Communications on Pure and Applied Analysis, 2020, 19 (1) : 541-585. doi: 10.3934/cpaa.2020027
[3]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
[4]

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

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413
[6]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
[7]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[8]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial and Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183
[9]

Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71
[10]

Danijela Damjanovic and Anatole Katok. Local rigidity of actions of higher rank abelian groups and KAM method. Electronic Research Announcements, 2004, 10: 142-154.

[11]

Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[12]

Ming Li, Shaobo Gan, Lan Wen. Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 239-269. doi: 10.3934/dcds.2005.13.239
[13]

Trygve K. Karper. Convergent finite differences for 1D viscous isentropic flow in Eulerian coordinates. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 993-1023. doi: 10.3934/dcdss.2014.7.993
[14]

Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693
[15]

Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545
[16]

Elvise Berchio, Debdip Ganguly. Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1871-1892. doi: 10.3934/cpaa.2016020
[17]

Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981
[18]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483
[19]

Tong Li, Sunčica Čanić. Critical thresholds in a quasilinear hyperbolic model of blood flow. Networks and Heterogeneous Media, 2009, 4 (3) : 527-536. doi: 10.3934/nhm.2009.4.527
[20]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (75)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy