Guido Gentile, M.V. Bartuccelli, J.H.B. Deane
Summation of divergent series and Borel summability 
for strongly dissipative equations 
with periodic or quasi-periodic forcing terms
(401K, postscript)

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.