# 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)
How to Cite This Entry:
Reducible linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reducible_linear_system&oldid=48464
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article