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Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities

Vieri Mastropietro·Michela Procesi
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Abstract

We prove the existence of small amplitude periodic solutions, for a large Lebesgue measure set of frequencies, in the nonlinear beam equation with a weak quadratic and velocity dependent nonlinearity and with Dirichlet boundary conditions. Such nonlinear PDE can be regarded as a simple model describing oscillations of flexible structures like suspension bridges in presence of an uniform wind flow. The periodic solutions are explicitly constructed by means of a perturbative expansion which can be considered the analogue of the Lindstedt series expansion for the invariant tori in classical mechanics. The periodic solutions are not analytic but defined only in a Cantor set, and resummation techniques of divergent powers series are used in order to control the small divisors problem.

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Vieri Mastropietro, Michela Procesi. 2005-05-13. Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities. https://arxiv.org/abs/math/0505283

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