Regular linear system

of ordinary differential equations

A system of the form

(1)

(where is a mapping that is summable on every interval and has the property that

exists and is equal to , where are the characteristic Lyapunov exponents (cf. Lyapunov characteristic exponent) of the system (1)).

For a triangular system

to be regular it is necessary and sufficient that the limits

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 of its characteristic Lyapunov exponents be negative:

Then for every system

(2)

where satisfies the following conditions: and are continuous, and , , where , there is a -dimensional manifold containing the point , such that every solution of (2) starting on (i.e. ) exponentially decreases as ; more precisely, for every there is a such that the inequality

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