- 06-110 Guido Gentile, Michele V. Bartuccelli, Jonathan H.B. Deane
- Quasi-periodic attractors, Borel summability and
the Bryuno condition for strongly dissipative systems
Apr 7, 06
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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.