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, andresummation techniques of divergent powers series are used in order to control the small divisors problem.

V. MASTROPIETRO, & M. PROCESI (2006). Lindstedt series for periodic solutions of beam equations under quadratic and velocity dependent nonlinearities. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 5, 1-28 [10.3934/cpaa.2006.5.1].

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

PROCESI, MICHELA
2006

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, andresummation techniques of divergent powers series are used in order to control the small divisors problem.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11590/139419
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