September  2006, 16(3): 635-655. doi: 10.3934/dcds.2006.16.635

On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition

1. 

Department of Mathematics, Southeast University, Nanjing 210096, China

2. 

Department of Mathematics, Southeast University, Nanjing 211189, China

Received  January 2006 Revised  June 2006 Published  August 2006

In this paper we prove the persistence of elliptic lower dimensional invariant tori for nearly integrable Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth with respect to parameters in the sense of Whitney, with a Gevrey index depending on the Gevrey class of Hamiltonian systems and on the exponent in the Diophantine condition. Moreover the Gevrey index should be optimal for the Diophantine condition in the proof of our theorem.
Citation: Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635
[1]

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

[2]

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

[3]

Claire Chavaudret, Stefano Marmi. Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition. Journal of Modern Dynamics, 2012, 6 (1) : 59-78. doi: 10.3934/jmd.2012.6.59

[4]

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

[5]

Pedro J. Torres, Zhibo Cheng, Jingli Ren. Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2155-2168. doi: 10.3934/dcds.2013.33.2155

[6]

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

[7]

Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587

[8]

Genni Fragnelli, Dimitri Mugnai. Singular parabolic equations with interior degeneracy and non smooth coefficients: The Neumann case. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-17. doi: 10.3934/dcdss.2020084

[9]

Claire Chavaudret, Stefano Marmi. Erratum: Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition. Journal of Modern Dynamics, 2015, 9: 285-287. doi: 10.3934/jmd.2015.9.285

[10]

Peter Giesl. Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 355-373. doi: 10.3934/dcds.2007.18.355

[11]

Nadia Lekrine, Chao-Jiang Xu. Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation. Kinetic & Related Models, 2009, 2 (4) : 647-666. doi: 10.3934/krm.2009.2.647

[12]

Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587

[13]

Peter Šepitka. Riccati equations for linear Hamiltonian systems without controllability condition. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1685-1730. doi: 10.3934/dcds.2019074

[14]

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

[15]

Sandra Lucente, Eugenio Montefusco. Non-hamiltonian Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 761-770. doi: 10.3934/dcdss.2013.6.761

[16]

Sonja Hohloch, Silvia Sabatini, Daniele Sepe. From compact semi-toric systems to Hamiltonian $S^1$-spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 247-281. doi: 10.3934/dcds.2015.35.247

[17]

Plamen Stefanov, Yang Yang. Multiwave tomography with reflectors: Landweber's iteration. Inverse Problems & Imaging, 2017, 11 (2) : 373-401. doi: 10.3934/ipi.2017018

[18]

Chiara Leone, Anna Verde, Giovanni Pisante. Higher integrability results for non smooth parabolic systems: The subquadratic case. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 177-190. doi: 10.3934/dcdsb.2009.11.177

[19]

Hongwei Lou, Junjie Wen, Yashan Xu. Time optimal control problems for some non-smooth systems. Mathematical Control & Related Fields, 2014, 4 (3) : 289-314. doi: 10.3934/mcrf.2014.4.289

[20]

Mikhail I. Belishev, Aleksei F. Vakulenko. Non-smooth unobservable states in control problem for the wave equation in $\mathbb{R}^3$. Evolution Equations & Control Theory, 2014, 3 (2) : 247-256. doi: 10.3934/eect.2014.3.247

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]