A Simple Proof of Gevrey Estimates for Expansions of Quasi-Periodic Orbits: Dissipative Models and Lower-Dimensional Tori

Abstract

We consider standard-like/Froeschlé dissipative maps with a dissipation and nonlinear perturbation. That is,

$$T_{\varepsilon}(p,q)=\left((1-\gamma\varepsilon^{3})p+\mu+\varepsilon V^{\prime}(q),q+(1-\gamma\varepsilon^{3})p+\mu+\varepsilon V^{\prime}(q)\bmod 2\pi\right)$$

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 \(\gamma\neq 0\). 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 \(\gamma\neq 0\), 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.