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 quasiperiodic 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 quasiperiodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasiperiodic 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 even the case of periodic forcing terms is nontrivial, as the solution is not analytic in a neighborhood of the origin: it turns out to be Borel summable. In the case of quasiperiodic 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; however, this cannot be interpreted as Borel summability.

Document Downloads