We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasiperiodic forcing term and in the presence of damping. In the limit of large damping, under some generic nondegeneracy condition on the force, there are quasiperiodic solutions which have the same frequency vector as the forcing term. We prove that such olutions 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 quasiperiodic 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 quasiperiodic solution is proved to describe a local attractor.

Gentile, G., BARTUCCELLI M., V., Deane, J.H.B. (2006). Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems. JOURNAL OF MATHEMATICAL PHYSICS, 47, 072702, 1-10 [10.1063/1.2213790].

Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems

GENTILE, Guido;
2006-01-01

Abstract

We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasiperiodic forcing term and in the presence of damping. In the limit of large damping, under some generic nondegeneracy condition on the force, there are quasiperiodic solutions which have the same frequency vector as the forcing term. We prove that such olutions 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 quasiperiodic 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 quasiperiodic solution is proved to describe a local attractor.
Gentile, G., BARTUCCELLI M., V., Deane, J.H.B. (2006). Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems. JOURNAL OF MATHEMATICAL PHYSICS, 47, 072702, 1-10 [10.1063/1.2213790].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/141562
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