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Special Issue: On the 80th birthday of professor A. Chenciner
A Simple Proof of Gevrey Estimates for Expansions of Quasi-Periodic Orbits: Dissipative Models and Lower-Dimensional Tori
Adrián P. Bustamantea,
Rafael de la Llaveb a Department of Mathematics, University of Roma Tor Vergata,
Via della Ricerca Scientifica 1, 00133 Roma, Italy
b School of Mathematics, Georgia Institute of Technology,
686 Cherry St., 30332-1160 Atlanta GA, USA
Abstract:
We consider standard-like/Froeschlé dissipative maps
with a dissipation and nonlinear perturbation. That is,
$$
T_\varepsilon(p,q) = \left(
(1 - \gamma \varepsilon^3) p + \mu + \varepsilon V'(q),
q + (1 - \gamma \varepsilon^3) p + \mu + \varepsilon V'(q) \bmod 2 \pi \right)
$$
where
$p \in {\mathbb R}^D$,
$q \in {\mathbb T}^D$ are the dynamical
variables. We fix a frequency
$\omega \in {\mathbb R}^D$ and study the existence of
quasi-periodic orbits. When there is dissipation, having
a quasi-periodic orbit of frequency
$\omega$ requires
selecting the parameter
$\mu$, called
the drift.
We first study the Lindstedt series (formal power series in
$\varepsilon$) for quasi-periodic orbits with
$D$ independent frequencies and the drift when
$\gamma \ne 0$.
We show that, when
$\omega$ is
irrational, the series exist to all orders, and when
$\omega$ 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
$\gamma = 0$ and when
$\gamma \ne 0$, 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.
Keywords:
Lindstedt series, Gevrey series, asymptotic expansions, resonances, whiskered tori.
MSC: 35C20,
34K26,
37J40,
70K43,
70K70 Received: 30.03.2023
Accepted: 07.09.2023
Language: English
DOI:
10.1134/S1560354723040123
References