Integral separation condition

A condition on a system of linear differential equations

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

(where $ A $ is a mapping $ \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $ with $ \sup _ {t \in \mathbf R } \| A ( t) \| < \infty $), requiring that the system has solutions $ x _ {i} ( t) $, $ i = 1 \dots n $, satisfying for certain $ a , d > 0 $ the inequalities

$$ | x _ {i} ( t) | \cdot | x _ {i} ( \tau ) | ^ {-} 1 \geq \ d e ^ {a ( t - \tau ) } \cdot | x _ {i-} 1 ( t) | \cdot | x _ {i-} 1 ( \tau ) | ^ {-} 1 $$

for all $ i = 2 \dots n $ and all $ t \geq \tau \geq 0 $.

The set of systems satisfying the integral separation condition is the interior of the set of continuity of all Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent) in the space of systems

$$ \dot{x} = A ( t) x ,\ \ \sup _ {x \in \mathbf R } \| A ( t) \| < + \infty , $$

with metric

$$ \rho ( A ( t) , B ( t) ) = \ \sup _ {t \in \mathbf R } \| A ( t) - B ( t) \| . $$

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