Abstract:
We consider standard-like/Froeschlé dissipative maps
with a dissipation and nonlinear perturbation. That is,
Tε(p,q)=((1−γε3)p+μ+εV′(q),q+(1−γε3)p+μ+εV′(q)mod2π)
where p∈RD, q∈TD are the dynamical
variables. We fix a frequency ω∈RD and study the existence of
quasi-periodic orbits. When there is dissipation, having
a quasi-periodic orbit of frequency ω requires
selecting the parameter μ, called the drift.
We first study the Lindstedt series (formal power series in ε) for quasi-periodic orbits with D independent frequencies and the drift when γ≠0.
We show that, when ω is
irrational, the series exist to all orders, and when ω is Diophantine,
we show that the formal Lindstedt series are Gevrey.
The Gevrey nature of the Lindstedt series above was shown
in [3] using a more general method, but the present proof is
rather elementary.
We also study the case when D=2, but the quasi-periodic orbits
have only one independent frequency (lower-dimensional tori).
Both when γ=0 and when γ≠0, we show
that, under some mild nondegeneracy conditions on V, there
are (at least two) formal Lindstedt series defined to all orders
and that they are Gevrey.
A. B. acknowledges the MIUR Excellence Department Project awarded to the Department
of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. A.B. was partially
supported by the MIUR-PRIN 20178CJA2B “New Frontiers of Celestial Mechanics: Theory and
Applications”.
Citation:
Adrián P. Bustamante, Rafael de la Llave, “A Simple Proof of Gevrey Estimates for Expansions of Quasi-Periodic Orbits: Dissipative Models and Lower-Dimensional Tori”, Regul. Chaotic Dyn., 28:4-5 (2023), 707–730
\Bibitem{BusDe 23}
\by Adri\'an P. Bustamante, Rafael de la Llave
\paper A Simple Proof of Gevrey Estimates for Expansions of Quasi-Periodic Orbits: Dissipative Models and Lower-Dimensional Tori
\jour Regul. Chaotic Dyn.
\yr 2023
\vol 28
\issue 4-5
\pages 707--730
\mathnet{http://mi.mathnet.ru/rcd1229}
\crossref{https://doi.org/10.1134/S1560354723040123}
Linking options:
https://www.mathnet.ru/eng/rcd1229
https://www.mathnet.ru/eng/rcd/v28/i4/p707
This publication is cited in the following 1 articles:
Adrián P. Bustamante, “Computation of domains of analyticity of lower dimensional tori in a weakly dissipative Froeschlé map”, Communications in Nonlinear Science and Numerical Simulation, 142 (2025), 108538