# American Institute of Mathematical Sciences

• Previous Article
Global attractor and rotation number of a class of nonlinear noisy oscillators
• DCDS Home
• This Issue
• Next Article
Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems
March  2007, 18(2&3): 569-595. doi: 10.3934/dcds.2007.18.569

## Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method

 1 Institute of Energy Problems of Chemical Physics, The Russia Academy of Sciences, Leninskiĭ prospect 38, Bldg. 2, Moscow 119334, Russian Federation

Received  December 2005 Revised  May 2006 Published  March 2007

We consider quasi-periodic (with $N$ basic frequencies) non-autonomous perturbations of Hamiltonian, reversible, volume preserving, and dissipative systems. The unperturbed systems possess analytic families of invariant $n$-tori carrying conditionally periodic motions, are allowed to depend on external parameters, and are assumed to satisfy just very weak nondegeneracy conditions. We construct invariant $(n+N)$-tori in perturbed systems following M.R. Herman's approach: additional external parameters are introduced to remove degeneracies and then are eliminated via an appropriate number-theoretical lemma concerning Diophantine approximations of dependent quantities.
Citation: Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569
 [1] Àlex Haro, Rafael de la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1261-1300. doi: 10.3934/dcdsb.2006.6.1261 [2] Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006 [3] Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537 [4] Xin Zhang, Shuangling Yang. Complex dynamics in a quasi-periodic plasma perturbations model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4013-4043. doi: 10.3934/dcdsb.2020272 [5] Ugo Locatelli, Letizia Stefanelli. Quasi-periodic motions in a special class of dynamical equations with dissipative effects: A pair of detection methods. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1155-1187. doi: 10.3934/dcdsb.2015.20.1155 [6] Zhichao Ma, Junxiang Xu. A KAM theorem for quasi-periodic non-twist mappings and its application. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3169-3185. doi: 10.3934/dcds.2022013 [7] Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467 [8] Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75 [9] Zhihua Ren, Tian Wang, Hao Wu. Comments on Poincaré theorem for quasi-periodic systems. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022115 [10] Bochao Chen, Yixian Gao. Quasi-periodic travelling waves for beam equations with damping on 3-dimensional rectangular tori. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 921-944. doi: 10.3934/dcdsb.2021075 [11] Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104 [12] Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585 [13] Junxiang Xu. On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2593-2619. doi: 10.3934/dcds.2013.33.2593 [14] Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9 [15] Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169 [16] Xavier Blanc, Claude Le Bris. Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Networks and Heterogeneous Media, 2010, 5 (1) : 1-29. doi: 10.3934/nhm.2010.5.1 [17] Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41 [18] Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75 [19] Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171 [20] Jinhao Liang. Positive Lyapunov exponent for a class of quasi-periodic cocycles. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1361-1387. doi: 10.3934/dcds.2020080

2021 Impact Factor: 1.588