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Moser's theorem with frequency-preserving

Part of: Low-dimensional dynamical systems Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  29 August 2023

Chang Liu ,
Zhicheng Tong Open the ORCID record for Zhicheng Tong [Opens in a new window]  and
Yong Li
Chang Liu
Affiliation:
School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, People's Republic of China (liuc947@nenu.edu.cn)
Zhicheng Tong
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, People's Republic of China (tongzc20@mails.jlu.edu.cn)
Yong Li
Affiliation:
Institute of Mathematics, Jilin University, Changchun 130012, People's Republic of China (liyong@jlu.edu.cn) School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024, People's Republic of China (liyong@jlu.edu.cn)
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Abstract

This paper mainly concerns the KAM persistence of the mapping $\mathscr {F}:\mathbb {T}^{n}\times E\rightarrow \mathbb {T}^{n}\times \mathbb {R}^{n}$ with intersection property, where $E\subset \mathbb {R}^{n}$ is a connected closed bounded domain with interior points. By assuming that the frequency mapping satisfies certain topological degree condition and weak convexity condition, we prove some Moser-type results about the invariant torus of mapping $\mathscr {F}$ with frequency-preserving under small perturbations. To our knowledge, this is the first approach to Moser's theorem with frequency-preserving. Moreover, given perturbed mappings over $\mathbb {T}^n$, it is shown that such persistence still holds when the frequency mapping and perturbations are only continuous about parameter beyond Lipschitz or even Hölder type. We also touch the parameter without dimension limitation problem under such settings.

Keywords

frequency-preservinginvariant torusmapping with intersection property

MSC classification

Primary: 37E40: Twist maps
Secondary: 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 31
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Arnol'd, V. I.. Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18 (1963), 9–36.CrossRefGoogle Scholar
Bounemoura, A.. Positive measure of KAM tori for finitely differentiable Hamiltonians. J. Éc. Polytech. Math. 7 (2020), 1113–1132.CrossRefGoogle Scholar
Broer, H. W., Huitema, G. B., Takens, F. and Braaksma, B. L. J.. Unfoldings and bifurcations of quasi-periodic tori. Mem. Am. Math. Soc. 83 (1990), viii+175.Google Scholar
Cheng, C. and Sun, Y.. Existence of invariant tori in three-dimensional measure-preserving mappings. Celestial Mech. Dyn. Astron. 47 (1989/90), 275–292.CrossRefGoogle Scholar
Cheng, C. and Sun, Y.. Existence of KAM tori in degenerate Hamiltonian systems. J. Differ. Equ. 114 (1994), 288–335.CrossRefGoogle Scholar
Chierchia, L. and Qian, D.. Moser's theorem for lower dimensional tori. J. Differ. Equ. 206 (2004), 55–93.CrossRefGoogle Scholar
Chow, S., Li, Y. and Yi, Y.. Persistence of invariant tori on submanifolds in Hamiltonian systems. J. Nonlinear Sci. 12 (2002), 585–617.CrossRefGoogle Scholar
Cong, F., Li, Y. and Huang, M.. Invariant tori for nearly twist mappings with intersection property. Northeast. Math. J. 12 (1996), 280–298.Google Scholar
Du, J., Li, Y. and Zhang, H.. Kolmogorov's theorem for degenerate Hamiltonian systems with continuous parameters. https://doi.org/10.48550/arXiv.2206.05461.CrossRefGoogle Scholar
Feng, K. and Shang, Z.. Volume-preserving algorithms for source-free dynamical systems. Numer. Math. 71 (1995), 451–463.Google Scholar
Herman, M. R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5–233.CrossRefGoogle Scholar
Herman, M. R.. Sur les courbes invariantes par les difféomorphismes de l'anneau I. Astérisque 103-104 (1983), 1–221.Google Scholar
Herman, M. R.. Sur les courbes invariantes par les difféomorphismes de l'anneau II. Astérisque 144 (1986), 248.Google Scholar
Katznelson, Y. and Ornstein, D.. The absolute continuity of the conjugation of certain diffeomorphisms of the circle. Ergod. Theory Dyn. Syst. 9 (1989), 681–690.CrossRefGoogle Scholar
Kolmogorov, A. N.. On conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl. Akad. Nauk SSSR 98 (1954), 527–530.Google Scholar
Koudjinan, C. E.. A KAM theorem for finitely differentiable Hamiltonian systems. J. Differ. Equ. 269 (2020), 4720–4750.CrossRefGoogle Scholar
Levi, M. and Moser, J.. A Lagrangian proof of the invariant curve theorem for twist mapping . Proc. Sympos. Pure Math. Am. Math. Soc., Providence, RI 69 (2001), 733–746.CrossRefGoogle Scholar
Li, Y. and Yi, Y.. Persistence of invariant tori in generalized Hamiltonian systems. Ergod. Theory Dyn. Syst. 22 (2002), 1233–1261.CrossRefGoogle Scholar
Li, Y. and Yi, Y.. Nekhoroshev and KAM stabilities in generalized Hamiltonian systems. J. Dyn. Differ. Equ. 18 (2006), 577–614.CrossRefGoogle Scholar
Liu, B.. Invariant curves of quasi-periodic reversible mappings. Nonlinearity 18 (2005), 685–701.CrossRefGoogle Scholar
Liu, C. and Xing, J.. A new proof of Moser's theorem. J. Appl. Anal. Comput. 12 (2022), 1679–1701.Google Scholar
Moser, J.. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. KI. II 1962 (1962), 1–20.Google Scholar
Moser, J.. A stability theorem for minimal foliations on a torus. Ergod. Theory Dyn. Syst. 8 (1988), 251–281.Google Scholar
Parasyuk, I. O.. Preservation of multidimensional invariant tori of Hamiltonian systems. Ukr. Mat. Zh. 36 (1984), 467–473.Google Scholar
Popov, G.. KAM theorem for Gevrey Hamiltonians. Ergod. Theory Dyn. Syst. 24 (2004), 1753–1786.CrossRefGoogle Scholar
Qian, W., Li, Y. and Yang, X.. Multiscale KAM theorem for Hamiltonian systems. J. Differ. Equ. 266 (2019), 70–86.CrossRefGoogle Scholar
Qian, W., Li, Y. and Yang, X.. Melnikov's conditions in matrices. J. Dyn. Differ. Equ. 32 (2020), 1779–1795.CrossRefGoogle Scholar
Rüssmann, H.. On the existence of invariant curves of twist mappings of an annulus. Geometric dynamics, 677-718. Lecture Notes in Math., vol. 1007 (Berlin: Springer, 1983).CrossRefGoogle Scholar
Rüssmann, H.. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Regul. Chaotic Dyn. 6 (2001), 119–204.CrossRefGoogle Scholar
Salamon, D. A.. The Kolmogorov-Arnold-Moser theorem. Math. Phys. Electron. J. 10 (2004), 37.Google Scholar
Sevryuk, M. B.. Reversible systems. Lecture Notes in Mathematics, vol. 1211, pp. vi+319 (Berlin: Springer-Verlag, 1986).CrossRefGoogle Scholar
Sevryuk, M. B.. KAM-stable Hamiltonians. J. Dyn. Control Syst. 1 (1995), 351–366.CrossRefGoogle Scholar
Shang, Z.. A note on the KAM theorem for symplectic mappings. J. Dyn. Differ. Equ. 12 (2000), 357–383.CrossRefGoogle Scholar
Siegel, C. L. and Moser, J. K.. Lectures on celestial mechanics, pp. xii+290 (New York-Heidelberg: Springer-Verlag, 1971).CrossRefGoogle Scholar
Svanidze, N. V.. Small perturbations of an integrable dynamical system with an integral invariant. Boundary value problems of mathematical physics, 10. Trudy Mat. Inst. Steklov. 147 (1980), 124–146.Google Scholar
Tong, Z., Du, J. and Li, Y.. KAM theorem on modulus of continuity about parameter. Sci. China Math. (2023). https://doi.org/10.1007/s11425-022-2102-5.CrossRefGoogle Scholar
Xia, Z.. Existence of invariant tori in volume-preserving diffeomorphisms. Ergod. Theory Dyn. Syst. 12 (1992), 621–631.CrossRefGoogle Scholar
Xu, J., You, J. and Qiu, Q.. Invariant tori for nearly integrable Hamiltonian systems with degeneracy. Math. Z. 226 (1997), 375–387.CrossRefGoogle Scholar
Yang, L. and Li, X.. Existence of periodically invariant tori on resonant surfaces for twist mappings. Discrete Contin. Dyn. Syst. 40 (2020), 1389–1409.CrossRefGoogle Scholar
Zhao, X. and Li, Y.. A Moser theorem for multiscale mappings. Discrete Contin. Dyn. Syst. 42 (2022), 3931–3951.CrossRefGoogle Scholar