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Volume 37 • Number 4 • 2014
• Linearization for Systems with Partially Hyperbolic Linear Part
Yong-Hui Xia

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.

2010 Mathematics Subject Classification: 34N05, 26E70, 39A21

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