Persistence of the non-twist torus in nearly integrable hamiltonian systems
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- by Junxiang Xu and Jiangong You PDF
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Abstract:
In this paper we consider analytic nearly integrable hamiltonian systems, and prove that if the frequency mapping has nonzero Brouwer topological degree at some Diophantine frequency, then the invariant torus with this frequency persists under small perturbations.References
- V. I. Arnol′d, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 13–40 (Russian). MR 0163025
- V. I. Arnol′d, V. V. Kozlov, and A. I. Neĭshtadt, Dynamical systems. III, Encyclopaedia of Mathematical Sciences, vol. 3, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. Iacob. MR 923953, DOI 10.1007/978-3-642-61551-1
- H. W. Broer, G. B. Huitema, F. Takens, and B. L. J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, Mem. Amer. Math. Soc. 83 (1990), no. 421, viii+175. MR 1041003, DOI 10.1090/memo/0421
- Chong-Qing Cheng, Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems, Comm. Math. Phys. 177 (1996), no. 3, 529–559. MR 1385075, DOI 10.1007/BF02099537
- L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 1, 115–147 (1989). MR 1001032
- A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.) 98 (1954), 527–530 (Russian). MR 0068687
- Jürgen Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136–176. MR 208078, DOI 10.1007/BF01399536
- J. Pöschel. A lecture on the classical KAM theorem. School on dynamical systems. May 1992.
- Jürgen Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35 (1982), no. 5, 653–696. MR 668410, DOI 10.1002/cpa.3160350504
- H. Rüssmann. On twist hamiltonian. Talk in the Colloque International: Mécanique Céleste et Systémes hamiltoniens, Marseille, 1990.
- Helmut Rüssmann, Nondegeneracy in the perturbation theory of integrable dynamical systems, Stochastics, algebra and analysis in classical and quantum dynamics (Marseille, 1988) Math. Appl., vol. 59, Kluwer Acad. Publ., Dordrecht, 1990, pp. 211–223. MR 1052709
- 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, KAM-stable Hamiltonians, J. Dynam. Control Systems 1 (1995), no. 3, 351–366. MR 1354540, DOI 10.1007/BF02269374
- 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
- Junxiang Xu and Jiangong You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann’s non-degeneracy condition, J. Differential Equations 235 (2007), no. 2, 609–622. MR 2317497, DOI 10.1016/j.jde.2006.12.001
- 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
Additional Information
- Junxiang Xu
- Affiliation: Department of Mathematics, Southeast University, Nanjing 210096, People’s Republic of China
- Email: xujun@seu.edu.cn
- Jiangong You
- Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
- MR Author ID: 241618
- Email: jyou@nju.edu.cn
- Received by editor(s): February 19, 2009
- Received by editor(s) in revised form: August 3, 2009
- Published electronically: February 18, 2010
- Additional Notes: The first author was supported by the National Natural Science Foundation of China (10571027)
The second author was partially supported by the National Basic Research Program of China (973 Program, 2007CB814800) and by the NNSF of China (Grant No. 10531050) - Communicated by: Yingfei Yi
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 2385-2395
- MSC (2010): Primary 34D10, 34D23; Secondary 34C27
- DOI: https://doi.org/10.1090/S0002-9939-10-10151-8
- MathSciNet review: 2607868