




Volume 37 • Number 4 • 2014 

•
Linearization for Systems with Partially Hyperbolic Linear Part
YongHui Xia
Abstract.
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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