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Antidotes for non-integrability of nonlinear systems: Quasi-periodic motions. (English) Zbl 0705.34047

Computational solution of nonlinear systems of equations, Proc. SIAM-AMS Summer Semin., Ft. Collins/CO (USA) 1988, Lect. Appl. Math. 26, 31-36 (1990).
[For the entire collection see Zbl 0688.00015.]
To solve a (Hamiltonian) ODE approximately, e.g., \(\ddot q-a\dot q-bq^ 3=h(t),\) where a, b are positive numbers, h(t) is quasi-periodic with frequencies \((\omega_ 1,...,\omega_ n)\), the author changes it via \(q(t)=u(\omega_ 1t,...,\omega_ nt)\) into a PDE for \(u(t_ 1,...,t_ n)\) for which a (variational) functional \(\Phi\) (u) is established and studied. He arrives at the (open) problem: Can the infimum of the functional \(\Phi\) (u) be used to compute quasi-periodic motions by optimization methods?
(Reviewers remark: Because of a, \(b>0\) the particular ODE is not related to a physical system.)
Reviewer: E.Brommundt

MSC:

34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
94A15 Information theory (general)

Citations:

Zbl 0688.00015