On a nonlinear perturbation theory without Secular Terms II. Carleman embedding of nonlinear equations in an infinite set of linear ones

1 January 1976

proceedings article
Published by AIP Publishing in AIP Conference Proceedings

Vol. 27 (1) , 75-110
https://doi.org/10.1063/1.30362

Abstract

C. Eminhizer and the authors1 have recently developed iterative methods for solving the dynamics of extremely nonlinear anharmonic oscillators. These methods are re‐examined here in a formalism proposed by Carleman2 for the embedding of nonlinear dynamical equations in an infinite set of linear equations. The methods are based on recursion formulae for the Fourier coefficients of Fourier series solutions of the dynamical equations and for coefficients of expansions of Fourier frequencies. The series solutions avoid ’’Secular Terms’’ and ’’Small Denominators’’ and converge rapidly in large regimes of highly nonlinear behaviour.

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