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 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 , 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.
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July 2006
Research Article|
July 18 2006
Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems
Guido Gentile;
Guido Gentile
a)
Dipartimento di Matematica,
Università di Roma Tre
, Roma, I-00146, Italy
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Michele V. Bartuccelli;
Michele V. Bartuccelli
b)
Department of Mathematics and Statistics,
University of Surrey
, Guildford, GU2 7XH, United Kingdom
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Jonathan H. B. Deane
Jonathan H. B. Deane
c)
Department of Mathematics and Statistics,
University of Surrey
, Guildford, GU2 7XH, United Kingdom
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J. Math. Phys. 47, 072702 (2006)
Article history
Received:
March 14 2006
Accepted:
May 23 2006
Citation
Guido Gentile, Michele V. Bartuccelli, Jonathan H. B. Deane; Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems. J. Math. Phys. 1 July 2006; 47 (7): 072702. https://doi.org/10.1063/1.2213790
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