# Regular linear system

of ordinary differential equations

A system of the form

$$\tag{1 } \dot{x} = A( t) x,\ x \in \mathbf R ^ {n}$$

(where $A( \cdot )$ is a mapping $\mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {m} )$ that is summable on every interval and has the property that

$$\lim\limits _ {t \rightarrow \infty } \frac{1}{t} \int\limits _ { 0 } ^ { t } \mathop{\rm Tr} A( \tau ) d \tau$$

exists and is equal to $\sum _ {i= 1 } ^ { n } \lambda _ {i} ( A)$, where $\lambda _ {1} ( A) \geq \dots \geq \lambda _ {n} ( A)$ are the characteristic Lyapunov exponents (cf. Lyapunov characteristic exponent) of the system (1)).

For a triangular system

$$\dot{u} ^ {i} = \sum _ {j= i } ^ { n } p _ {ij} ( t) u ^ {j} ,\ i= 1 \dots n,$$

to be regular it is necessary and sufficient that the limits

$$\lim\limits _ {t \rightarrow \infty } \frac{1}{t} \int\limits _ { 0 } ^ { t } p _ {ii} ( \tau ) d \tau ,\ i= 1 \dots n,$$

exist (Lyapunov's criterion). Every reducible linear system and every almost-reducible linear system is regular.

The role of the concept of a regular linear system is clarified by the following theorem of Lyapunov. Let the system (1) be regular and let $k$ of its characteristic Lyapunov exponents be negative:

$$0 > \lambda _ {n-} k+ 1 ( A) \geq \dots \geq \lambda _ {n} ( A).$$

Then for every system

$$\tag{2 } \dot{x} = A( t) x + g( t, x),$$

where $g( t, x)$ satisfies the following conditions: $g$ and $g _ {x} ^ \prime$ are continuous, and $g( t, 0)= 0$, $\sup _ {t \geq 0 } \| g _ {x} ^ \prime ( t, x) \| = O( | x | ^ \epsilon )$, where $\epsilon = \textrm{ const } > 0$, there is a $k$- dimensional manifold $V ^ {k} \subset \mathbf R ^ {n}$ containing the point $x= 0$, such that every solution $x( t)$ of (2) starting on $V ^ {k}$( i.e. $x( 0) \in V ^ {k}$) exponentially decreases as $t \rightarrow \infty$; more precisely, for every $\delta > 0$ there is a $C _ \delta$ such that the inequality

$$| x( t) | \leq C _ \delta e ^ {[ \lambda _ {n-} k+ 1 ( A)+ \delta ] t } | x( 0) |$$

is satisfied.

#### References

 [1] A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian) [2] B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow (1966) (In Russian) [3] 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 : 1 (1974) pp. 71–146
How to Cite This Entry:
Regular linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_linear_system&oldid=48482
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article