Uniform methods to establish Poincaré type linearization theorems. (English) Zbl 1326.34067

Wu, Hao

Publ. Mat., Barc. 2014, Extra, 497-527 (2014).

This work deals with differential systems of the form \[ \frac{dx}{dt}= \Lambda x + f(t,x), \] where \(x\) is \(d\)-dimensional, \(t \in \mathbb{R}\), \(\Lambda\) is a constant matrix in Jordan normal form and \(f(t,x)\) is continuous in \(t\) and analytic in \(x\) at \(x=0\) for every fixed \(t\), \(f(t,0) \equiv 0\) and \(D_x f(t,0) \equiv 0\). The author establishes four theorems which belong to the domain of normal form theory. In the first theorem, using majorant norm method, a set of conditions to linearize the differential system is established. One of the main hypothesis in Theorem 1 is that \(\Lambda\) is a diagonal matrix. The second theorem is related to the case in which the function \(f\) does not depend on \(t\), that is the system is autonomous. Then, also conditions for linearizability are obtained.
At this point, the author considers random dynamical systems and establishes a result for its linearizability under several conditions. Last but not least the author applies Theorem 1 to systems for which averaging theory can be used. In this way, he considers families of dynamical systems depending on a parameter \(\varepsilon\) and establishes when there exists a change of coordinates (also depending on \(\varepsilon\)) for which the system can be written in a particular form, that is, the one needed to apply averaging theory.
The paper provides the complete proof of the four Theorems as well as several applications of them. This paper is specially interesting for researchers on the topic of normal form theory and its applications. The results are very original.

Reviewer: Maite Grau (Lleida)

MSC:

34C20	Transformation and reduction of ordinary differential equations and systems, normal forms
37H10	Generation, random and stochastic difference and differential equations

Keywords:

normal form; random dynamical system; almost periodic system; majorant norm method