A simple proof of Gevrey estimates for expansions of quasi-periodic orbits: dissipative models and lower dimensional tori

Abstract

We consider standard-like/Froeschlé maps with a dissipation and nonlinear perturbation. That is \[ T_\varepsilon(p,q) = \left( (1 - γ\varepsilon^3) p + μ+ \varepsilon V'(q), q + (1 - γ\varepsilon^3) p + μ+ \varepsilon V'(q) \mod 2 π\right) \] where $p\in {\mathbb{R}}^D$, $q\in 2\pi {\mathbb{T}}^D$ are the dynamical variables. The $\mu \in {\mathbb{R}}^D,\gamma \in \mathbb{R}$ are parameters of the model. We assume that the potential $V$ is a trigonometric polynomial. Note that when $\gamma \ne 0$, the perturbation parameter $\epsilon$ creates dissipation, which has a drastic effect on the existence of quasi-periodic orbits, hence it is a singular perturbation. We fix a frequency $\omega \in {\mathbb{R}}^D$ and study the existence of quasiperiodic orbits. When there is dissipation, having a quasiperiodic orbit of frequency $\omega$ requires adjusting the parameter $\mu$, called \textit{the drift}. We first study the Lindstedt series (formal power series in $\epsilon$) for quasiperidic 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. 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 non-degeneracy conditions on $V$, there are (at least two) formal Lindstedt series defined to all orders and that they are Gevrey. Furthermore, we can take $\mu$ along a $1$ dimensional space.