Linearization for Systems with Partially Hyperbolic Linear Part

Yong-Hui Xia

Bulletin of the Malaysian Mathematical Society, Tome 37 (2014) no. 4

Résumé

To study the linearization problem of dynamic system on measure chains (time scales), the authors in the previous work assumed that linear system $x^{\Delta}=A(t)x$ should possess exponential dichotomy. In this paper, the assumption is weakened and the setting on the whole linear part $x^{\Delta}=A(t)x$ need not to be hyperbolic. We only need assume partially hyperbolic linear part. More specifically, if system $x^{\Delta}=A(t)x$ is rewritten as two subsystems $ \left\{\begin{array}{lll} x_1^{\Delta}=A_1(t)x_1, \\ x_2^{\Delta}=A_2(t)x_2 \end{array}\right. $, it requires that the first subsystem $x_1^{\Delta}=A_1(t)x_1$ has exponential dichotomy, while there is no requirement on the other linear subsystem $x_2^{\Delta}=A_2(t)x_2$. That is, the whole linear system $x^{\Delta}=A(t)x$ need not to possess exponential dichotomy. The previous result is improved in this paper.

Détail

Classification : 34N05, 26E70, 39A21