You, Jiangong Invariant tori and Lagrange stability of pendulum-type equations. (English) Zbl 0702.34047 J. Differ. Equations 85, No. 1, 54-65 (1990). This paper is motivated by a question put by Moser in 1973 concerning the Lagrange stability of non-autonomous pendulum-like equations of the form \(x'=y\), \(y'=-G_ x(t,x)+p(t)\) where G and p have period 1. It is shown that such a smooth system is Lagrangian stable iff p has mean zero. It has an infinite number of invariant tori if p has mean zero, and none otherwise. Reviewer: J.F.Toland Cited in 24 Documents MSC: 34D20 Stability of solutions to ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 37C75 Stability theory for smooth dynamical systems 34C25 Periodic solutions to ordinary differential equations Keywords:Lagrange stability; non-autonomous pendulum-like equations PDFBibTeX XMLCite \textit{J. You}, J. Differ. Equations 85, No. 1, 54--65 (1990; Zbl 0702.34047) Full Text: DOI References: [1] Moser, J., Stable and random motions in dynamical systems, Ann. of Math. Stud., 77 (1973) [2] Mawhin, J.; Willem, M., Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52, 264-287 (1984) · Zbl 0557.34036 [3] Willem, M., Oscillations forcées de systèmes hamiltoniens, (Publications Séminaires Analyse non linéaire de l’Univ. de Besançon (1981)) · Zbl 0482.70020 [4] Zehnder, E., An “a priori” estimate for oscillatory equations, (Lecture Notes in Mathematics, Vol. 1125 (1983), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0585.34028 [5] Moser, J., On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1-20 (1962) · Zbl 0107.29301 [6] M. Levi, KAM theory for particles in periodic potential, Ergodic Theory Dynamical Systems, in press.; M. Levi, KAM theory for particles in periodic potential, Ergodic Theory Dynamical Systems, in press. [7] Moser, J., Minimal foliations on a torus, Ergodic Theory Dynamical Systems, 8 (1988) · Zbl 0632.57018 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.