FM 2006-4 (mp_arc 06-110; arXiv math.DS/0604162

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

Title: Quasi-periodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems
 
Abstract: We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasi-periodic forcing term and in the presence of damping. In the limit of large damping, under some generic non-degeneracy condition on the force, there are quasi-periodic solutions which have the same frequency vector as the forcing term. We prove that such solutions are Borel summable at the origin when the frequency vector is either any one-dimensional number or a two-dimensional vector such that the ratio of its components is an irrational number of constant type. In the first case the proof given simplifies that provided in a previous work of ours. We also show that in any dimension $d$, for the existence of a quasi-periodic solution with the same frequency vector as the forcing term, the standard Diophantine condition can be weakened into the Bryuno condition. In all cases, under a suitable positivity condition, the quasi-periodic solution is proved to describe a local attractor.

Keywords: Dissipative systems; Periodically forced systems; Quasi-periodically forced systems; Lindstedt series; Renormalization group; Divergent series; Borel summability; Bryuno vectors; Irrational numbers of constant type.

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