FM 2005-3 (mp_arc 05-36; arXiv math.DS/0501500

Author: Guido Gentile, Michele V. Bartuccelli, Jonathan H.B. Deane

Title: Summation of divergent series and Borel summability for strongly dissipative equations with periodic or quasi-periodic forcing terms
 
Abstract: We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasi-periodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistor-inductor-varactor circuit with a periodic (or quasi-periodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasi-periodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties in the inverse of the damping coefficient. We find that already the case of periodic forcing terms is non-trivial, as the solution is not analytic in a neighbourhood of the origin: it turns out to be Borel-summable. In the case of quasi-periodic forcing terms we need Renormalization Group techniques in order to control the small divisors arising in the perturbation series. We show the existence of a summation criterion of the series in this case also, but, however, this can not be interpreted as Borel summability.

Keywords: Dissipative systems; Periodically forced systems; Quasi-periodically forced systems; Lindstedt series; Renormalization group; Divergent series; Borel summability.

Guido Gentile
Dipartimento di Matematica
Università di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@mat.uniroma3.it

Michele Bartuccelli
Department of Mathematics and Statistics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: m.bartuccelli@surrey.ac.uk

Jonathan Deane
Department of Mathematics and Statistics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: j.deane@surrey.ac.uk