Bin, Liu Boundedness for solutions of nonlinear Hill’s equations with periodic forcing terms via Moser’s twist theorem. (English) Zbl 0675.34024 J. Differ. Equations 79, No. 2, 304-315 (1989). The author considers the Hill’s equation with periodic forcing term \[ (*)\quad x''+\beta x^{2n+1}+(a_ 1+\epsilon a(t))x=p(t),\quad n\geq 1, \] where a(t) and p(t) are continuous and 1-periodic functions in \(t\in R\), \(a_ 1\) and \(\beta\) are positive constants and \(\epsilon\) is a small parameter. He proves that there exists an \(\epsilon_ 0\) such that for a ll \(\epsilon \in (0,\epsilon_ 0)\) every solution to the equation (*) is bounded together with its derivative. Under some assumptions there exists quasi-periodic solution or m-periodic solution for every integer \(m\geq 1\). Reviewer: D.Bobrowski Cited in 44 Documents MSC: 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations Keywords:time-independent Hamiltonian system; Moser’s twist theory; Hill’s equation; periodic forcing term; small parameter PDFBibTeX XMLCite \textit{L. Bin}, J. Differ. Equations 79, No. 2, 304--315 (1989; Zbl 0675.34024) Full Text: DOI References: [1] Arscott, F. M., Periodic Differential Equations (1964), Pergamon Press: Pergamon Press New York · Zbl 0121.29903 [2] Cesari, L., Asymptotic Behavior and Stability Problems (1963), Academic Press: Academic Press New York · Zbl 0111.08701 [3] Magnus, W.; Winkler, S., Hill’s Equation (1966), Interscience: Interscience New York · Zbl 0158.09604 [4] Dieckerhoff, R.; Zehnder, E., (Boundedness of solutions via twist theorem, Vol. 22 (1984), Abteilung für Math. der Rühr-Univ: Abteilung für Math. der Rühr-Univ Bochum), preprint · Zbl 0656.34027 [5] Ding, T., Boundedness of solutions of Duffing’s equations, J. Differential Equations, 61, No. 2, 178-207 (1986) · Zbl 0619.34038 [7] Herman, M., Introduction à l’étude des courbes invariantes par les difféomorphismes de l’anneau (1983), Asterisque [8] Moser, J., On invariant curves of area-preserving mapping of annulus, Nachr. Akad. Wiss. Göttingen Math.-phys. Kl. II, 1-20 (1962) · Zbl 0107.29301 [9] Birkhoff, G. D., Proof of Poincaré’s geometric theorem, Amer. Math. Soc., 14, 14-22 (1913) · JFM 44.0761.01 [10] Morris, G., A case of boundedness in Littlewood’s problem on oscillatory differential equations, Bull. Austral. Math. Soc., 14, 71-93 (1976) · Zbl 0324.34030 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.