View source for Irregularity indices

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''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|regular linear system]].

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

1) The Lyapunov irregularity index [[#References|[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|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 [[#References|[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|variational equations]] of a [[Hamiltonian system|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 [[#References|[4]]]–.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,   "Collected works" , '''2''' , Moscow-Leningrad  (1956)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Perron,   "Die Ordnungszahlen linearer Differentialgleichungssysteme"  ''Math. Z.'' , '''31'''  (1929–1930)  pp. 748–766</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.G. Malkin,   "Theorie der Stabilität einer Bewegung" , R. Oldenbourg , München  (1959)  pp. Sect. 79  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.V. Nemytskii,   "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top">  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</TD></TR></table>

====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} $.