**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