**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