×

Boundedness of solutions for Duffing’s equations with semilinear potentials. (English) Zbl 1004.34020

The paper concerns the equation \(\ddot{x}(t)+\arctan x(t)=\varepsilon e(t)\) where \(e\in C^\infty(S^1)\) and \(\varepsilon\) is a small parameter. The main result states that every solution \(x\) to the equation is defined on \(\mathbb{R}\), and both \(x\) and \(\dot{x}\) are bounded there. The proof method indicates the same conclusion for the more general equation \(\ddot{x}(t)+g(x(t))=\varepsilon e(t)\) where \(g\in C^\infty(\mathbb{R})\), \(x g(x)>0\) for \(x\neq 0\), and further definite hypotheses are satisfied.

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Arnold, V. I., Mathematical Methods of Classical Mechanics (1978), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0386.70001
[2] Dieckerhoff, R.; Zehnder, E., Boundedness of solutions via the twist theorem, Ann. Scuola. Norm. Sup. Pisa Cl. Sci., 14, 79-95 (1987) · Zbl 0656.34027
[3] Ding, T., Nonlinear oscillations at a point of resonance, Scientia Sinica, 25, 918-931 (1982) · Zbl 0509.34043
[4] Herman, M., Sur les courbes invariantes par les diffemorphismes de l’anneau I, Asterisque, 103-104 (1983)
[5] Kupper, T.; You, J., Existence of quasiperiodic solutions and Littlewood’s boundedness problem of Duffing equations with subquadratic potentials, Nonlinear Anal., 35, 549-559 (1999) · Zbl 0927.34021
[6] Laederich, S.; Levi, M., Invariant curves and time-dependent potentials, Ergodic Theory Dynam. Systems, 11, 365-378 (1991) · Zbl 0742.34050
[7] Levi, M., Quasi-periodic motions in superquadratic time-periodic potentials, Comm. Math. Phys., 143, 43-83 (1991) · Zbl 0744.34043
[8] Littlewood, J., Some Problems in Real and Complex Analysis (1968), Heath: Heath Lexington · Zbl 0185.11502
[9] B. Liu, On Littlewood’s boundedness problem for sublinear Duffing equations, preprint.; B. Liu, On Littlewood’s boundedness problem for sublinear Duffing equations, preprint. · Zbl 0974.34031
[10] Liu, B., Boundedness for solutions of nonlinear Hill’s equations with periodic forcing terms via Moser’s twist theorem, J. Differential Equations, 79, 304-315 (1989) · Zbl 0675.34024
[11] Liu, B., Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 355-373 (1999) · Zbl 0926.34026
[12] Liu, B., Boundedness of solutions for semilinear Duffing’s equation, J. Differential Equations, 145, 119-144 (1998) · Zbl 0913.34032
[13] Liu, B., Boundedness in nonlinear oscillations at resonance, J. Differential Equations, 153, 142-172 (1999) · Zbl 0926.34028
[14] B. Liu, and, F. Zanolin, Boundedness of solutions for second order quasilinear ODEs, preprint.; B. Liu, and, F. Zanolin, Boundedness of solutions for second order quasilinear ODEs, preprint.
[15] Mather, J., Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, 21, 457-467 (1982) · Zbl 0506.58032
[16] Massera, J. L., The existence of periodic solutions of system of differential equations, Duke Math. J., 17, 457-475 (1950) · Zbl 0038.25002
[17] Morris, G., A case of boundedness of Littlewood’s problem on oscillatory differential equations, Bull. Austral. Math. Soc., 14, 71-93 (1976) · Zbl 0324.34030
[18] Moser, J., Stable and Random Motions in Dynamical System (1973), Princeton Univ. Press: Princeton Univ. Press Princeton
[19] Moser, J., On invariant curves of area-preserving mappings of annulus, Nachr. Akad. Wiss. Gottingen Math.-Phys., kl.II, 1-20 (1962) · Zbl 0107.29301
[20] Ortega, R., Asymmetric oscillators and twist mappings, J. London Math. Soc., 53, 325-342 (1996) · Zbl 0860.34017
[21] R. Ortega, Invariant curves of mappings with averaged small twist, preprint.; R. Ortega, Invariant curves of mappings with averaged small twist, preprint. · Zbl 1136.37336
[22] R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc, in press.; R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc, in press. · Zbl 1030.34035
[23] R. Ortega, An report on the boundedness for semilinear Duffing’s equation, talk, Institute of Mathematics, Academia Sinica, 1998.; R. Ortega, An report on the boundedness for semilinear Duffing’s equation, talk, Institute of Mathematics, Academia Sinica, 1998.
[24] Pei, M., Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations, J. Differential Equations, 113, 106-127 (1994) · Zbl 0808.34045
[25] Russman, H., On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Mathematics (1981), Springer-verlag: Springer-verlag Berlin/Heidelberg/New York, p. 677-718
[26] Wang, Y.; You, J., Boundedness of solutions in polynomial potentials with \(C^2\) cofficients, Z. Angew. Math. Phys., 47, 943-952 (1996) · Zbl 0957.34034
[27] Wang, Y., KAM Theorem and Lagrangian Stability for Nonlinear Oscillations (1999), Peking University
[28] You, J., Boundedness for solutions of superlinear Duffing’s equtions via twist curves theorems, Scientia Sinica, 35, 399-412 (1992)
[29] Yuan, X., Invariant tori of Duffing-type equations, J. Differential Equations, 142, 231-262 (1998) · Zbl 0911.34042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.