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

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  • 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 Dirichelet 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 a convergent perturbative expansion which can be considered the analogue of the Lindstedt series expansion for the invariant tori in classical mechanics. The periodic solutions are defined only in a Cantor set, and resummation techniques of divergent powers series are used in order to control the small divisors problem.
    Mathematics Subject Classification: Primary 35B10, 35B32, 35L70, 47H15, 47N20, 58F27, 58F39.


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