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