Normal form and linearization for quasiperiodic systems

Abstract:

In this paper, we consider the following system of differential equations: \[ \dot \theta = \omega + \Theta (\theta ,z), \quad \dot z = Az + f(\theta ,z),\] where $\theta \in {C^m}$, $\omega = ({\omega _1}, \ldots ,{\omega _m}) \in {R^m}$, $z \in {C^n}$, $A$ is a diagonalizable matrix, $f$ and $\Theta$ are analytic functions in both variables and $2\pi$-periodic in each component of the vector $\theta ,\Theta = O(|z|)$ and $f = O(|z{|^2})$ as $z \to 0$. We study the normal form of this system of the equations and prove that this system can be transformed to a system of linear equations \[ \dot \theta = \omega , \quad \dot z = Az\] by an analytic transformation provided that the eigenvalues of $A$ and the frequency $\omega$ satisfy certain small-divisor conditions.