Irregularity indices

for linear systems of ordinary differential equations

Non-negative functions $ \sigma $ on the space of mappings $ A : \mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $ (or $ \mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf C ^ {n} , \mathbf C ^ {n} ) $), integrable on every finite interval, such that $ \sigma ( A ) $ equals zero if and only if the system

$$ \tag{* } \dot{x} = A ( t) x $$

is a regular linear system.

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

1) The Lyapunov irregularity index [1]:

$$ \sigma _ {L} ( A ) = \ \sum _ { i= 1} ^ { n } \lambda _ {i} ( A ) - \lim\limits _ {\overline{ {t \rightarrow + \infty }}\; } \ \frac{1}{t} \int\limits _ { 0 } ^ { t } \mathop{\rm tr} A ( \tau ) d \tau , $$

where $ \lambda _ {i} ( A) $ are the Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent) of the system (*), arranged in descending order, while $ \mathop{\rm tr} A ( t) $ is the trace of the mapping $ A ( t) $.

2) The Perron irregularity index [2]:

$$ \sigma _ {p} ( A) = \ \max _ {1 \leq i \leq n } ( \lambda _ {i} ( A) + \lambda _ {n+ 1- i} ( - A ^ {*} ) ) , $$

where $ A ^ {*} ( t) $ is the adjoint of the mapping $ A ( t) $. If the system (*) is a system of variational equations of a Hamiltonian system

$$ \dot{q} = \frac{\partial H }{\partial p } ,\ p \in \mathbf R ^ {k} , $$

$$ \dot{p} = - \frac{\partial H }{\partial q } ,\ q \in \mathbf R ^ {k} , $$

then $ n = 2k $ and

$$ \lambda _ {i} ( - A ^ {*} ) = \ \lambda _ {i} ( A ) ,\ \ i = 1 \dots n . $$

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

$$ \lambda _ {i} ( A ) = \ - \lambda _ {n+ 1}- i ( A) ,\ \ i = 1, \dots, k , $$

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 $ A : \mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf C ^ {n} , \mathbf C ^ {n} ) $, read $ \sum _ {i= 1} ^ {n} \mathop{\rm Re} A _ {ii} ( t) $ instead of $ \mathop{\rm tr} A ( t) $ in the definition of $ \sigma _ {L} $.