Lagrange Stability and KAM Tori for Duffing Equations with Quasi-periodic Coefficients

Abstract

It is proved that there are many (positive Lebesgue measure) Kolmogorov-Arnold-Moser (KAM for short) tori at infinity and thus all solutions are bounded for the Duffing equations

\ddot{x}+x^{2n+1}+\sum _{j=0}^{2n}{p}_i(t)x^j=0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\ddot x}+x^{2n+1}+\sum_{j=0}^{2n}p_{i}(t)x^{j}=0$$\end{document}

with p_j(t)’s being time-quasi-periodic smooth functions.

Keywords

KAM tori / Lagrangian stability / Duffing equation / Quasi-periodic function

References

[1] MoserJStable and Random Motions in Dynamical Systems: With special emphasis on celestial mechanics, 1973, Princeton, Princeton University Press

[2] KolmogorovA N. On conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk SSSR (N.S.), 1954, 98: 527-530

[3] Arnol’dV 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., 1963, 18(5): 9-36

[4] MoserJ. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Gottingen Math. Phys. KI. II, 1962, 1962: 1-20

[5] MorrisG. A case of boundedness of Littlewoods problem on oscillatory differential equations. Bull. Austral. Math. Soc., 1976, 14: 71-93

[6] DieckerhoffR, ZehnderE. Boundedness of solutions via twist theorem. Ann. Scula. Norm. Sup. Pisa., 1987, 14: 79-95

[7] LeviM. Quasi-periodic motions in superquadratic time-periodic potentials. Comm. Math. Phys., 1991, 143: 43-83

[8] LiuB. Boundedness for solutions of nonlinear Hill’s equations with periodic forcing terms via Moser’s twist theorem. J. Diff. Eqs., 1989, 79: 304-315

[9] LiuB. Boundedness for solutions of nonlinear periodic differential equations via Moser’s twist theorem. Acta Math. Sinica (N.S.), 1992, 8: 91-98

[10] YuanX P. Invariant Tori of Duffing-type Equations. Advances in Math. (China), 1995, 24: 375-376

[11] YuanX P. Invariant Tori of Duffiiig-type Equations. J. Differential Equations, 1998, 142(2): 231-262

[12] KuksinS, PöschelJ. Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrdinger equation. Ann. of Math., 1996, 143(1): 149-179 2)

[13] LiuB, YouJ. KAM tori for guasiperiodic forced differential equations. Proceedings of the Conference on Qualitative Theory of ODE Nanjing Daxue Xuebao Shuxue Bannian Kan, 199323-26(in Chinese)

[14] WuY, YuanX P. On the Kolmogorov theorem for some infinite-dimensional Hamiltonian systems of short range. Nonlinear Anal., 2021, 202: 112120 34 pp

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