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Invariant tori and Lagrange stability of pendulum-type equations. (English) Zbl 0702.34047

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

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
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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
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