Persistence of Floquet invariant tori for a class of non-conservative dynamical systems
HTML articles powered by AMS MathViewer
- by Junxiang Xu PDF
- Proc. Amer. Math. Soc. 135 (2007), 805-814 Request permission
Abstract:
In this paper we consider a class of non-conservative dynamical system with small perturbation. By the KAM method we prove existence of Floquet invariant tori under the weakest non-resonant conditions.References
- Jürgen Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136–176. MR 208078, DOI 10.1007/BF01399536
- Hendrik W. Broer, George B. Huitema, and Mikhail B. Sevryuk, Quasi-periodic motions in families of dynamical systems, Lecture Notes in Mathematics, vol. 1645, Springer-Verlag, Berlin, 1996. Order amidst chaos. MR 1484969
- Dario Bambusi and Giuseppe Gaeta, Invariant tori for non conservative perturbations of integrable systems, NoDEA Nonlinear Differential Equations Appl. 8 (2001), no. 1, 99–116. MR 1828951, DOI 10.1007/PL00001441
- Samuel M. Graff, On the conservation of hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations 15 (1974), 1–69. MR 365626, DOI 10.1016/0022-0396(74)90086-2
- Bin Liu, On lower dimensional invariant tori in reversible systems, J. Differential Equations 176 (2001), no. 1, 158–194. MR 1861186, DOI 10.1006/jdeq.2000.3960
- Jürgen Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z. 202 (1989), no. 4, 559–608. MR 1022821, DOI 10.1007/BF01221590
- H. Rüssmann, On twist-Hamiltonian. Talk held on the Colloque international: Mécanique céleste et systèmes hamiltoniens, Marseille, 1990.
- H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn. 6 (2001), no. 2, 119–204. MR 1843664, DOI 10.1070/RD2001v006n02ABEH000169
- M. B. Sevryuk, Reversible systems, Lecture Notes in Mathematics, vol. 1211, Springer-Verlag, Berlin, 1986. MR 871875, DOI 10.1007/BFb0075877
- M. B. Sevryuk, Invariant $m$-dimensional tori of reversible systems with a phase space of dimension greater than $2m$, Trudy Sem. Petrovsk. 14 (1989), 109–124, 266–267 (Russian, with English summary); English transl., J. Soviet Math. 51 (1990), no. 3, 2374–2386. MR 1001357, DOI 10.1007/BF01094996
- Chong Qing Cheng and Yi Sui Sun, Existence of KAM tori in degenerate Hamiltonian systems, J. Differential Equations 114 (1994), no. 1, 288–335. MR 1302146, DOI 10.1006/jdeq.1994.1152
- Junxiang Xu, Jiangong You, and Qingjiu Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z. 226 (1997), no. 3, 375–387. MR 1483538, DOI 10.1007/PL00004344
- Junxiang Xu and Jiangong You, Persistence of lower-dimensional tori under the first Melnikov’s non-resonance condition, J. Math. Pures Appl. (9) 80 (2001), no. 10, 1045–1067. MR 1876763, DOI 10.1016/S0021-7824(01)01221-1
- Junxiang Xu, Normal form of reversible systems and persistence of lower dimensional tori under weaker nonresonance conditions, SIAM J. Math. Anal. 36 (2004), no. 1, 233–255. MR 2083860, DOI 10.1137/S0036141003421923
- Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3
Additional Information
- Junxiang Xu
- Affiliation: Department of Mathematics, Southeast University, Nanjing 210096, People’s Republic of China
- Email: xujun@seu.edu.cn
- Received by editor(s): February 19, 2004
- Received by editor(s) in revised form: October 15, 2005
- Published electronically: September 11, 2006
- Additional Notes: This paper is Project 10571027 supported by the NSFC
- Communicated by: Carmen C. Chicone
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 805-814
- MSC (2000): Primary 34D20; Secondary 34C05
- DOI: https://doi.org/10.1090/S0002-9939-06-08529-7
- MathSciNet review: 2262876