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.