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''of a linear system of ordinary differential equations''
 
''of a linear system of ordinary differential equations''
  
 
The quantities defined by:
 
The quantities defined by:
  
<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/s/s085/s085540/s0855401.png" /></td> </tr></table>
+
$$
 +
\Omega  ^ {0} ( A)  = \overline{\lim\limits}\; _ {\theta - \tau \rightarrow + \infty } \
 +
 
 +
\frac{1}{\theta - \tau }
 +
  \mathop{\rm ln}  \| X( \theta , \tau ) \|
 +
$$
  
 
(the upper singular exponent) and
 
(the upper singular exponent) 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/s/s085/s085540/s0855402.png" /></td> </tr></table>
+
$$
 +
\omega  ^ {0} ( A)  = \lim\limits _ {\theta - \overline{ {\tau \rightarrow + }}\; \infty } \
 +
 
 +
\frac{1}{\tau - \theta }
 +
  \mathop{\rm ln}  \| X( \tau , \theta ) \|
 +
$$
  
(the lower singular exponent), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s0855403.png" /> is the [[Cauchy operator|Cauchy operator]] (i.e. the fundamental solution or principal solution) of the system
+
(the lower singular exponent), where $  X( \theta , \tau ) $
 +
is the [[Cauchy operator|Cauchy operator]] (i.e. the fundamental solution or principal solution) of 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/s/s085/s085540/s0855404.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\dot{x}  = A( t) x,\  x \in \mathbf R  ^ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s0855405.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s0855406.png" /> that is summable on every interval.
+
where $  A( \cdot ) $
 +
is a mapping $  \mathbf R  ^ {+} \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $
 +
that is summable on every interval.
  
The singular exponents can be equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s0855407.png" />; if for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s0855408.png" />,
+
The singular exponents can be equal to $  \pm  \infty $;  
 +
if for a certain $  T > 0 $,
  
<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/s/s085/s085540/s0855409.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</td></tr></table>
+
$$ \tag{1'}
 +
\sup _ {t \in \mathbf R  ^ {+} }  \int\limits _ { t } ^ { t+ }  T \| A( \tau ) \|  d \tau  < + \infty ,
 +
$$
  
 
then the singular exponents are numbers.
 
then the singular exponents are numbers.
  
For a system (1) with constant coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554010.png" />, the singular exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554012.png" /> are equal to, respectively, the maximum and minimum of the real parts of the eigenvalues of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554013.png" />. For a system (1) with periodic coefficients (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554015.png" /> for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554016.png" />), the singular exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554018.png" /> are equal to, respectively, the maximum and minimum of the logarithms of the absolute values of the [[Multipliers|multipliers]], divided by the period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554019.png" />. The singular exponents are sometimes also called general exponents (see [[#References|[4]]]).
+
For a system (1) with constant coefficients $  ( A( t) \equiv A( 0)) $,  
 +
the singular exponents $  \Omega  ^ {0} ( A) $
 +
and $  \omega  ^ {0} ( A) $
 +
are equal to, respectively, the maximum and minimum of the real parts of the eigenvalues of the operator $  A( 0) $.  
 +
For a system (1) with periodic coefficients ( $  A( t+ T) = A( t) $
 +
for all $  t \in \mathbf R $
 +
for a certain $  T > 0 $),  
 +
the singular exponents $  \Omega  ^ {0} ( A) $
 +
and $  \omega  ^ {0} ( A) $
 +
are equal to, respectively, the maximum and minimum of the logarithms of the absolute values of the [[Multipliers|multipliers]], divided by the period $  T $.  
 +
The singular exponents are sometimes also called general exponents (see [[#References|[4]]]).
  
The following definitions are equivalent to those mentioned above if (1prm) holds for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554020.png" />: The singular exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554021.png" /> is equal to the greatest lower bound of the set of those numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554022.png" /> for each of which there is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554023.png" /> such that for any solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554024.png" /> of the system (1) the inequality
+
The following definitions are equivalent to those mentioned above if (1'}) holds for a certain $  T > 0 $:  
 +
The singular exponent $  \Omega  ^ {0} ( A) $
 +
is equal to the greatest lower bound of the set of those numbers $  \alpha $
 +
for each of which there is a number $  C _  \alpha  > 0 $
 +
such that for any solution $  x( t) \neq 0 $
 +
of the system (1) the inequality
  
<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/s/s085/s085540/s08554025.png" /></td> </tr></table>
+
$$
 +
| x( \theta ) |  \leq  C _  \alpha  e ^ {\alpha ( \theta - \tau ) } | x( \tau )
 +
| \  \textrm{ for }  \textrm{ all }  \theta \geq  \tau \geq  0
 +
$$
  
is fulfilled; the singular exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554026.png" /> is equal to the least upper bound of the set of those numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554027.png" /> for each of which a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554028.png" /> exists such that for every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554029.png" /> of the system (1) the inequality
+
is fulfilled; the singular exponent $  \omega  ^ {0} ( A) $
 +
is equal to the least upper bound of the set of those numbers $  \beta $
 +
for each of which a number $  C _  \beta  > 0 $
 +
exists such that for every solution $  x( t) \neq 0 $
 +
of the system (1) the inequality
  
<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/s/s085/s085540/s08554030.png" /></td> </tr></table>
+
$$
 +
| x( \theta ) |  \geq  C _  \beta  e ^ {\beta ( \theta - \tau ) }
 +
| x( \tau ) | \  \textrm{ for }  \textrm{ all }  \theta \geq  \tau \geq  0
 +
$$
  
 
is fulfilled.
 
is fulfilled.
  
For the singular exponents and for the Lyapunov characteristic exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]), for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554031.png" /> the inequalities
+
For the singular exponents and for the Lyapunov characteristic exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]), for each $  T > 0 $
 +
the inequalities
  
<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/s/s085/s085540/s08554032.png" /></td> </tr></table>
+
$$
 +
\sup _ {t \in \mathbf R  ^ {+} }
 +
 +
\frac{1}{T}
 +
\int\limits _ { t } ^ { t+ }  T \| A( \tau ) \|  d
 +
\tau  \geq  \Omega  ^ {0} ( A)  \geq  \lambda _ {1} ( A)  \geq  \dots
 +
$$
  
<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/s/s085/s085540/s08554033.png" /></td> </tr></table>
+
$$
 +
\dots \geq  \lambda _ {n} ( A)  \geq  \omega  ^ {0} ( A) \
 +
\geq  - \sup _ {t \in \mathbf R  ^ {+} } 
 +
\frac{1}{T}
 +
\int\limits _ { t } ^ { t+ }  T \| A( \tau ) \|  d \tau
 +
$$
  
 
hold. For linear systems with constant or periodic coefficients,
 
hold. For linear systems with constant or periodic coefficients,
  
<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/s/s085/s085540/s08554034.png" /></td> </tr></table>
+
$$
 +
\Omega  ^ {0} ( A)  = \lambda _ {1} ( A),\  \omega  ^ {0} ( A)  = \lambda _ {n} ( A),
 +
$$
  
 
but there exist systems for which the corresponding inequalities are strict (see [[Uniform stability|Uniform stability]]).
 
but there exist systems for which the corresponding inequalities are strict (see [[Uniform stability|Uniform stability]]).
  
The singular exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554035.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554036.png" />), as a function on the space of all systems of the form (1) with bounded continuous coefficients (the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554037.png" /> is continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554038.png" />) provided with the metric
+
The singular exponent $  \Omega  ^ {0} ( A) $(
 +
respectively, $  \omega  ^ {0} ( A) $),  
 +
as a function on the space of all systems of the form (1) with bounded continuous coefficients (the mapping $  A( \cdot ) $
 +
is continuous and $  \sup _ {t \in \mathbf R  ^ {+}  }  \| A( t) \| < + \infty $)  
 +
provided with the metric
  
<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/s/s085/s085540/s08554039.png" /></td> </tr></table>
+
$$
 +
d( A, B)  = \sup _ {t \in \mathbf R  ^ {+} }  \| A( t) - B( t) \| ,
 +
$$
  
 
is semi-continuous from above (respectively, from below) but is not continuous everywhere.
 
is semi-continuous from above (respectively, from below) but is not continuous everywhere.
  
If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554040.png" /> is uniformly continuous and
+
If the mapping $  A : \mathbf R \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $
 +
is uniformly continuous 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/s/s085/s085540/s08554041.png" /></td> </tr></table>
+
$$
 +
\sup _ {t \in \mathbf R }  \| A( t) \|  < + \infty ,
 +
$$
  
then the [[Shift dynamical system|shift dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554042.png" /> has invariant normalized measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554044.png" /> concentrated on the closure of the trajectory of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554045.png" /> such that, for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554046.png" /> (in the sense of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554047.png" />), the upper singular exponent of the system
+
then the [[Shift dynamical system|shift dynamical system]] $  ( S = \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} )) $
 +
has invariant normalized measures $  \mu _ {1} $
 +
and $  \mu _ {2} $
 +
concentrated on the closure of the trajectory of the point $  A $
 +
such that, for almost all $  \widetilde{A}  $(
 +
in the sense of the measure $  \mu _ {1} $),  
 +
the upper singular exponent of 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/s/s085/s085540/s08554048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\dot{x}  = \widetilde{A}  ( t) x
 +
$$
  
 
is equal to its largest (leading) Lyapunov characteristic exponent,
 
is equal to its largest (leading) Lyapunov characteristic exponent,
  
<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/s/s085/s085540/s08554049.png" /></td> </tr></table>
+
$$
 +
\Omega  ^ {0} ( \widetilde{A}  )  = \lambda _ {1} ( \widetilde{A}  ) ,
 +
$$
  
and for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554050.png" /> (in the sense of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554051.png" />), the lower singular exponent of the system (2) is equal to its smallest Lyapunov characteristic exponent,
+
and for almost all $  \widetilde{A}  $(
 +
in the sense of the measure $  \mu _ {2} $),  
 +
the lower singular exponent of the system (2) is equal to its smallest Lyapunov characteristic exponent,
  
<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/s/s085/s085540/s08554052.png" /></td> </tr></table>
+
$$
 +
\omega  ^ {0} ( \widetilde{A}  )  = \lambda _ {n} ( \widetilde{A}  ) .
 +
$$
  
For almost-periodic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554053.png" /> (see [[Linear system of differential equations with almost-periodic coefficients|Linear system of differential equations with almost-periodic coefficients]]) the measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554055.png" /> are identical and coincide with the unique normalized invariant measure concentrated on the restriction of the shift dynamical system to the closure of the trajectory of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554056.png" />, which in this case exists.
+
For almost-periodic mappings $  A( \cdot ) $(
 +
see [[Linear system of differential equations with almost-periodic coefficients|Linear system of differential equations with almost-periodic coefficients]]) the measures $  \mu _ {1} $
 +
and $  \mu _ {2} $
 +
are identical and coincide with the unique normalized invariant measure concentrated on the restriction of the shift dynamical system to the closure of the trajectory of the point $  A $,  
 +
which in this case exists.
  
Let a dynamical system on a smooth, closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554057.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554058.png" /> be defined by a smooth vector field. Then there exist normalized invariant measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554060.png" /> for this system such that for almost every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554061.png" /> (in the sense of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554062.png" />) the upper singular exponent and the leading Lyapunov characteristic exponent of the system of variational equations (equations in variations, linearized equations) along the trajectory of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554063.png" /> coincide, and for almost every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554064.png" /> (in the sense of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554065.png" />) the lower singular exponent and the smallest Lyapunov characteristic exponent of the system of variational equations along the trajectory of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554066.png" /> coincide. The definitions of singular exponents, Lyapunov characteristic exponents, etc., retain their meaning for systems of variational equations of smooth dynamical systems defined on arbitrary smooth manifolds. The system of variational equations of such a dynamical system along the trajectory of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554067.png" /> can be written in the form (1) by using, for example, that basis in the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554068.png" /> at every point of the trajectory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554069.png" /> which is obtained by a parallel transfer along this trajectory (in the sense of the Riemannian connection induced by the smooth Riemannian metric) of some basis of the tangent space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554070.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085540/s08554071.png" />.
+
Let a dynamical system on a smooth, closed $  n $-
 +
dimensional manifold $  V  ^ {n} $
 +
be defined by a smooth vector field. Then there exist normalized invariant measures $  \mu _ {1} $
 +
and $  \mu _ {2} $
 +
for this system such that for almost every point $  x \in V  ^ {n} $(
 +
in the sense of the measure $  \mu _ {1} $)  
 +
the upper singular exponent and the leading Lyapunov characteristic exponent of the system of variational equations (equations in variations, linearized equations) along the trajectory of the point $  x $
 +
coincide, and for almost every point $  x \in V  ^ {n} $(
 +
in the sense of the measure $  \mu _ {2} $)  
 +
the lower singular exponent and the smallest Lyapunov characteristic exponent of the system of variational equations along the trajectory of the point $  x $
 +
coincide. The definitions of singular exponents, Lyapunov characteristic exponents, etc., retain their meaning for systems of variational equations of smooth dynamical systems defined on arbitrary smooth manifolds. The system of variational equations of such a dynamical system along the trajectory of a point $  x $
 +
can be written in the form (1) by using, for example, that basis in the tangent space to $  V  ^ {n} $
 +
at every point of the trajectory of $  x $
 +
which is obtained by a parallel transfer along this trajectory (in the sense of the Riemannian connection induced by the smooth Riemannian metric) of some basis of the tangent space of $  V  ^ {n} $
 +
at $  x $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Bohl,  "Ueber Differentialgleichungen"  ''J. Reine Angew. Math.'' , '''144'''  (1913)  pp. 284–318</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Persidskii,  "First approximation kinetic stability"  ''Mat. Sb.'' , '''40''' :  3  (1933)  pp. 284–293  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.F. Bylov,  R.E. Vinograd,  D.M. Grobman,  V.V. Nemytskii,  "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.L. Daletskii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from 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  (1974)  pp. 46–96  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''12'''  (1974)  pp. 71–146</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Bohl,  "Ueber Differentialgleichungen"  ''J. Reine Angew. Math.'' , '''144'''  (1913)  pp. 284–318</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Persidskii,  "First approximation kinetic stability"  ''Mat. Sb.'' , '''40''' :  3  (1933)  pp. 284–293  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.F. Bylov,  R.E. Vinograd,  D.M. Grobman,  V.V. Nemytskii,  "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.L. Daletskii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from 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  (1974)  pp. 46–96  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''12'''  (1974)  pp. 71–146</TD></TR></table>

Latest revision as of 08:14, 6 June 2020


of a linear system of ordinary differential equations

The quantities defined by:

$$ \Omega ^ {0} ( A) = \overline{\lim\limits}\; _ {\theta - \tau \rightarrow + \infty } \ \frac{1}{\theta - \tau } \mathop{\rm ln} \| X( \theta , \tau ) \| $$

(the upper singular exponent) and

$$ \omega ^ {0} ( A) = \lim\limits _ {\theta - \overline{ {\tau \rightarrow + }}\; \infty } \ \frac{1}{\tau - \theta } \mathop{\rm ln} \| X( \tau , \theta ) \| $$

(the lower singular exponent), where $ X( \theta , \tau ) $ is the Cauchy operator (i.e. the fundamental solution or principal solution) of the system

$$ \tag{1 } \dot{x} = A( t) x,\ x \in \mathbf R ^ {n} , $$

where $ A( \cdot ) $ is a mapping $ \mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $ that is summable on every interval.

The singular exponents can be equal to $ \pm \infty $; if for a certain $ T > 0 $,

$$ \tag{1'} \sup _ {t \in \mathbf R ^ {+} } \int\limits _ { t } ^ { t+ } T \| A( \tau ) \| d \tau < + \infty , $$

then the singular exponents are numbers.

For a system (1) with constant coefficients $ ( A( t) \equiv A( 0)) $, the singular exponents $ \Omega ^ {0} ( A) $ and $ \omega ^ {0} ( A) $ are equal to, respectively, the maximum and minimum of the real parts of the eigenvalues of the operator $ A( 0) $. For a system (1) with periodic coefficients ( $ A( t+ T) = A( t) $ for all $ t \in \mathbf R $ for a certain $ T > 0 $), the singular exponents $ \Omega ^ {0} ( A) $ and $ \omega ^ {0} ( A) $ are equal to, respectively, the maximum and minimum of the logarithms of the absolute values of the multipliers, divided by the period $ T $. The singular exponents are sometimes also called general exponents (see [4]).

The following definitions are equivalent to those mentioned above if (1'}) holds for a certain $ T > 0 $: The singular exponent $ \Omega ^ {0} ( A) $ is equal to the greatest lower bound of the set of those numbers $ \alpha $ for each of which there is a number $ C _ \alpha > 0 $ such that for any solution $ x( t) \neq 0 $ of the system (1) the inequality

$$ | x( \theta ) | \leq C _ \alpha e ^ {\alpha ( \theta - \tau ) } | x( \tau ) | \ \textrm{ for } \textrm{ all } \theta \geq \tau \geq 0 $$

is fulfilled; the singular exponent $ \omega ^ {0} ( A) $ is equal to the least upper bound of the set of those numbers $ \beta $ for each of which a number $ C _ \beta > 0 $ exists such that for every solution $ x( t) \neq 0 $ of the system (1) the inequality

$$ | x( \theta ) | \geq C _ \beta e ^ {\beta ( \theta - \tau ) } | x( \tau ) | \ \textrm{ for } \textrm{ all } \theta \geq \tau \geq 0 $$

is fulfilled.

For the singular exponents and for the Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent), for each $ T > 0 $ the inequalities

$$ \sup _ {t \in \mathbf R ^ {+} } \frac{1}{T} \int\limits _ { t } ^ { t+ } T \| A( \tau ) \| d \tau \geq \Omega ^ {0} ( A) \geq \lambda _ {1} ( A) \geq \dots $$

$$ \dots \geq \lambda _ {n} ( A) \geq \omega ^ {0} ( A) \ \geq - \sup _ {t \in \mathbf R ^ {+} } \frac{1}{T} \int\limits _ { t } ^ { t+ } T \| A( \tau ) \| d \tau $$

hold. For linear systems with constant or periodic coefficients,

$$ \Omega ^ {0} ( A) = \lambda _ {1} ( A),\ \omega ^ {0} ( A) = \lambda _ {n} ( A), $$

but there exist systems for which the corresponding inequalities are strict (see Uniform stability).

The singular exponent $ \Omega ^ {0} ( A) $( respectively, $ \omega ^ {0} ( A) $), as a function on the space of all systems of the form (1) with bounded continuous coefficients (the mapping $ A( \cdot ) $ is continuous and $ \sup _ {t \in \mathbf R ^ {+} } \| A( t) \| < + \infty $) provided with the metric

$$ d( A, B) = \sup _ {t \in \mathbf R ^ {+} } \| A( t) - B( t) \| , $$

is semi-continuous from above (respectively, from below) but is not continuous everywhere.

If the mapping $ A : \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $ is uniformly continuous and

$$ \sup _ {t \in \mathbf R } \| A( t) \| < + \infty , $$

then the shift dynamical system $ ( S = \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} )) $ has invariant normalized measures $ \mu _ {1} $ and $ \mu _ {2} $ concentrated on the closure of the trajectory of the point $ A $ such that, for almost all $ \widetilde{A} $( in the sense of the measure $ \mu _ {1} $), the upper singular exponent of the system

$$ \tag{2 } \dot{x} = \widetilde{A} ( t) x $$

is equal to its largest (leading) Lyapunov characteristic exponent,

$$ \Omega ^ {0} ( \widetilde{A} ) = \lambda _ {1} ( \widetilde{A} ) , $$

and for almost all $ \widetilde{A} $( in the sense of the measure $ \mu _ {2} $), the lower singular exponent of the system (2) is equal to its smallest Lyapunov characteristic exponent,

$$ \omega ^ {0} ( \widetilde{A} ) = \lambda _ {n} ( \widetilde{A} ) . $$

For almost-periodic mappings $ A( \cdot ) $( see Linear system of differential equations with almost-periodic coefficients) the measures $ \mu _ {1} $ and $ \mu _ {2} $ are identical and coincide with the unique normalized invariant measure concentrated on the restriction of the shift dynamical system to the closure of the trajectory of the point $ A $, which in this case exists.

Let a dynamical system on a smooth, closed $ n $- dimensional manifold $ V ^ {n} $ be defined by a smooth vector field. Then there exist normalized invariant measures $ \mu _ {1} $ and $ \mu _ {2} $ for this system such that for almost every point $ x \in V ^ {n} $( in the sense of the measure $ \mu _ {1} $) the upper singular exponent and the leading Lyapunov characteristic exponent of the system of variational equations (equations in variations, linearized equations) along the trajectory of the point $ x $ coincide, and for almost every point $ x \in V ^ {n} $( in the sense of the measure $ \mu _ {2} $) the lower singular exponent and the smallest Lyapunov characteristic exponent of the system of variational equations along the trajectory of the point $ x $ coincide. The definitions of singular exponents, Lyapunov characteristic exponents, etc., retain their meaning for systems of variational equations of smooth dynamical systems defined on arbitrary smooth manifolds. The system of variational equations of such a dynamical system along the trajectory of a point $ x $ can be written in the form (1) by using, for example, that basis in the tangent space to $ V ^ {n} $ at every point of the trajectory of $ x $ which is obtained by a parallel transfer along this trajectory (in the sense of the Riemannian connection induced by the smooth Riemannian metric) of some basis of the tangent space of $ V ^ {n} $ at $ x $.

References

[1] P. Bohl, "Ueber Differentialgleichungen" J. Reine Angew. Math. , 144 (1913) pp. 284–318
[2] K. Persidskii, "First approximation kinetic stability" Mat. Sb. , 40 : 3 (1933) pp. 284–293 (In Russian)
[3] B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow (1966) (In Russian)
[4] Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)
[5] N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 : 1 (1974) pp. 46–96 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 71–146
How to Cite This Entry:
Singular exponents. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singular_exponents&oldid=48717
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article