# 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

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

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

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 $\sigma _ {L}$.