Integral separation condition

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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
How to Cite This Entry:
Integral separation condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_separation_condition&oldid=47382
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article