Normal fundamental system of solutions

of a linear homogeneous system of ordinary differential equations

A fundamental system of solutions $ x _ {1} ( t) \dots x _ {n} ( t) $ such that any other fundamental system $ \widehat{x} _ {1} ( t) \dots \widehat{x} _ {n} ( t) $ satisfies the inequality

$$ \sum _ { i= } 1 ^ { n } \lambda _ {\widehat{x} _ {i ( t) } } \geq \ \sum _ { i= } 1 ^ { n } \lambda _ {x _ {i ( t) } } ; $$

here

$$ \lambda _ {y ( t) } = \ \overline{\lim\limits}\; _ {t \rightarrow + \infty } \frac{1}{t} \mathop{\rm log} | y ( t) | $$

is the Lyapunov characteristic exponent of a solution $ y ( t) $. Normal fundamental systems of solutions were introduced by A.M. Lyapunov [1], who proved that they exist for every linear system

$$ \dot{x} = A ( t) x , $$

where $ A ( \cdot ) $ is a mapping

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

that is summable on every segment and satisfies the additional condition

$$ \overline{\lim\limits}\; _ {t \rightarrow \infty } \ \frac{1}{t} \int\limits _ { 0 } ^ { t } \| A ( \tau ) \| dt < + \infty . $$

References

[1]	A.M. Lyapunov, "Collected works" , 1–5 , Moscow-Leningrad (1956) (In Russian)