Irregularity indices

for linear systems of ordinary differential equations

Non-negative functions on the space of mappings (or ), integrable on every finite interval, such that equals zero if and only if the system

(*)

is a regular linear system.

The best known (and easiest to define) such regularity indices are as follows.

1) The Lyapunov irregularity index [1]:

where are the Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent) of the system (*), arranged in descending order, while is the trace of the mapping .

2) The Perron irregularity index [2]:

where is the adjoint of the mapping . If the system (*) is a system of variational equations of a Hamiltonian system

then and

Consequently, for a system of variational equations of a Hamiltonian system,

is a necessary and sufficient condition for regularity (a theorem of Persidskii).

For other irregularity indices, see [4]–.

References

[1]	A.M. Lyapunov, "Collected works" , 2 , Moscow-Leningrad (1956) (In Russian)
[2]	O. Perron, "Die Ordnungszahlen linearer Differentialgleichungssysteme" Math. Z. , 31 (1929–1930) pp. 748–766
[3]	I.G. Malkin, "Theorie der Stabilität einer Bewegung" , R. Oldenbourg , München (1959) pp. Sect. 79 (Translated from Russian)
[4]	B.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow (1966) (In Russian)
[5]	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
[6a]	R.A. Prokhorova, "Estimate of the jump of the highest exponent of a linear system due to exponential perturbations" Differential Eq. , 12 : 3 (1977) pp. 333–338 Differentsial'nye Uravneniya , 12 : 3 (1976) pp. 475–483
[6b]	R.A. Prokhorova, "Stability with respect to a first approximation" Differential Eq. , 12 : 4 (1977) pp. 539–542 Differentsial'nye Uravneniya , 12 : 4 (1976) pp. 766–796

Comments

In the case of , read instead of in the definition of .