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 |
Regular linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_linear_system&oldid=12155