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

This is the authors' abstract. We don't add key points for this paper.

We consider standard-like/Froeschlé dissipative maps with a dissipation and nonlinear perturbation. That is, Tε(p,q)=((1-γε³)p+μ+ε V^(q),q+(1-γε³)p+μ+ε V^(q) 2π) 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.