October  2010, 14(3): 1237-1249. doi: 10.3934/dcdsb.2010.14.1237

Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems

1. 

Faculty of mathematics and physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, China

2. 

Department of Mathematics, Southeast University, Nanjing 210096

Received  June 2009 Revised  December 2009 Published  July 2010

In this paper we consider the persistence of lower dimensional elliptic invariant tori with prescribed frequencies in reversible systems, and prove that if the frequency mapping has non-zero Brouwer's degree at a certain point that satisfies Melnikov's non-resonance conditions, then the invariant torus with given frequency persists under small perturbations.
Citation: Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1237-1249. doi: 10.3934/dcdsb.2010.14.1237
[1]

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

[2]

Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092

[3]

Shengqing Hu, Bin Liu. Degenerate lower dimensional invariant tori in reversible system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3735-3763. doi: 10.3934/dcds.2018162

[4]

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

[5]

Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683

[6]

Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. On the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1677-1692. doi: 10.3934/dcds.2016.36.1677

[7]

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

[8]

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

[9]

Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941

[10]

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

[11]

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

[12]

Paul H. Rabinowitz. On a class of reversible elliptic systems. Networks & Heterogeneous Media, 2012, 7 (4) : 927-939. doi: 10.3934/nhm.2012.7.927

[13]

Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433

[14]

Ugo Locatelli, Antonio Giorgilli. Invariant tori in the Sun--Jupiter--Saturn system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 377-398. doi: 10.3934/dcdsb.2007.7.377

[15]

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

[16]

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

[17]

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

[18]

Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123

[19]

Gemma Huguet, Rafael de la Llave, Yannick Sire. Computation of whiskered invariant tori and their associated manifolds: New fast algorithms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1309-1353. doi: 10.3934/dcds.2012.32.1309

[20]

Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]