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''for linear systems of ordinary differential equations''
 
''for linear systems of ordinary differential equations''
  
Non-negative functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i0527001.png" /> on the space of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i0527002.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i0527003.png" />), integrable on every finite interval, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i0527004.png" /> equals zero if and only if the system
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i0527005.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\dot{x}  = A ( t) x
 +
$$
  
 
is a [[Regular linear system|regular linear system]].
 
is a [[Regular linear system|regular linear system]].
Line 11: Line 29:
 
1) The Lyapunov irregularity index [[#References|[1]]]:
 
1) The Lyapunov irregularity index [[#References|[1]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i0527006.png" /></td> </tr></table>
+
$$
 +
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i0527007.png" /> are the Lyapunov characteristic exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]) of the system (*), arranged in descending order, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i0527008.png" /> is the trace of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i0527009.png" />.
+
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]]]:
 
2) The Perron irregularity index [[#References|[2]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270010.png" /></td> </tr></table>
+
$$
 +
\sigma _ {p} ( A)  = \
 +
\max _ {1 \leq  i \leq  n }
 +
( \lambda _ {i} ( A) + \lambda _ {n+} 1- i ( - A  ^ {*} ) ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270011.png" /> is the adjoint of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270012.png" />. If the system (*) is a system of [[Variational equations|variational equations]] of a [[Hamiltonian system|Hamiltonian system]]
+
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]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270013.png" /></td> </tr></table>
+
$$
 +
\dot{q}  =
 +
\frac{\partial  H }{\partial  p }
 +
,\  p \in \mathbf R  ^ {k} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270014.png" /></td> </tr></table>
+
$$
 +
\dot{p}  = -  
 +
\frac{\partial  H }{\partial  q }
 +
,\  q \in \mathbf R  ^ {k} ,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270015.png" /> and
+
then $  n = 2k $
 +
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270016.png" /></td> </tr></table>
+
$$
 +
\lambda _ {i} ( - A  ^ {*} )  = \
 +
\lambda _ {i} ( A ) ,\ \
 +
i = 1 \dots n .
 +
$$
  
 
Consequently, for a system of variational equations of a Hamiltonian system,
 
Consequently, for a system of variational equations of a Hamiltonian system,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270017.png" /></td> </tr></table>
+
$$
 +
\lambda _ {i} ( A )  = \
 +
- \lambda _ {n+} 1- i ( A) ,\ \
 +
i = 1 \dots k ,
 +
$$
  
 
is a necessary and sufficient condition for regularity (a theorem of Persidskii).
 
is a necessary and sufficient condition for regularity (a theorem of Persidskii).
Line 39: Line 92:
 
====References====
 
====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>
 
<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====
 
====Comments====
In the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270018.png" />, read <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270019.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270020.png" /> in the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052700/i05270021.png" />.
+
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} $.

Revision as of 22:13, 5 June 2020


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} $.

How to Cite This Entry:
Irregularity indices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irregularity_indices&oldid=47435
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article