Almost-reducible linear system

of ordinary differential equations

A system

$$ \tag{* } \dot{x} = A (t) x ,\ \ x \in \mathbf R ^ {n} , $$

$$ A ( \cdot ) : \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) , $$

having the following property: There exist a system $ \dot{y} = B y $, $ y \in \mathbf R ^ {n} $, with constant coefficients and, for every $ \epsilon > 0 $, a Lyapunov transformation $ L _ \epsilon (t) $ such that by the change of variables $ x = L _ \epsilon (t) y $, the system (*) is transformed into the system

$$ \dot{y} = ( B + C _ \epsilon (t) ) y , $$

where

$$ \sup _ {t \in \mathbf R } \ \| C _ \epsilon (t) \| < \epsilon . $$

Every reducible linear system is almost reducible.

References

[1]	N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 : 1 (1976) pp. 46–96 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 71–146