Reducible linear system

of ordinary differential equations

A system

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

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

that can be transformed into a system $ \dot{y} = By $ with constant coefficients by a change of variables $ x = L ( t) y $, where $ L ( t) $ is a Lyapunov transformation. If the mapping $ A ( t) $ is continuous and periodically depends on $ t $, then (*) is a reducible system (Lyapunov's theorem). The system (*) is reducible if and only if there is a Lyapunov transformation $ L ( t) $ and an operator $ B $ such that every solution of (*) has the form

$$ x ( t) = L ( t) e ^ {tB } x ( 0) $$

(Erugin's criterion).

References

[1]	A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[2]	N.P. Erugin, "Reducible systems" Trudy Mat. Inst. Steklov. , 13 (1946) (In Russian)