Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms
- 1 June 2005
- journal article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 46 (6)
- https://doi.org/10.1063/1.1926208
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\ud 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\ud 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 summabilityThis publication has 9 references indexed in Scilit:
- Quasi-Periodic Solutions for Two-Level SystemsCommunications in Mathematical Physics, 2003
- Quasianalytic monogenic solutions of a cohomological equationMemoirs of the American Mathematical Society, 2003
- Hyperbolic Low-Dimensional Invariant Tori¶and Summations of Divergent SeriesCommunications in Mathematical Physics, 2002
- Scaling properties for the radius of convergence of Lindstedt series: Generalized standard mapsJournal de Mathématiques Pures et Appliquées, 2000
- Limit at resonances of linearizations of some complex analytic dynamical systemsErgodic Theory and Dynamical Systems, 2000
- Scaling properties for the radius of convergence of a lindstedt series: the standard MAPJournal de Mathématiques Pures et Appliquées, 1999
- The distributional Borel summability and the large coupling ?4 lattice fieldsCommunications in Mathematical Physics, 1987
- The distributional Borel summability and the large coupling ?4 lattice fieldsCommunications in Mathematical Physics, 1986
- An improvement of Watson’s theorem on Borel summabilityJournal of Mathematical Physics, 1980