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''of a linear system of ordinary differential equations''
 
''of a linear system of ordinary differential equations''
  
 
Quantities defined by the formulas
 
Quantities defined by the formulas
  
<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/c/c021/c021170/c0211701.png" /></td> </tr></table>
+
$$
 +
\Omega ( A)  = \
 +
\lim\limits _ {T \rightarrow + \infty } \
 +
\overline{\lim\limits}\; _ {k \rightarrow + \infty } \
 +
{
 +
\frac{1}{kT}
 +
}
 +
\sum _ {i = 0 } ^ { {k }  - 1 }
 +
\mathop{\rm ln}  \| X (( i + 1) T, iT) \|
 +
$$
  
 
(the upper central exponent) and
 
(the upper central 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/c/c021/c021170/c0211702.png" /></td> </tr></table>
+
$$
 +
\omega ( A)  = \
 +
\lim\limits _ {T \rightarrow + \infty } \
 +
\overline{\lim\limits}\; _ {k \rightarrow + \infty } \
 +
 
 +
\frac{- 1 }{kT }
 +
 
 +
\sum _ {i = 0 } ^ { {k }  - 1 }
 +
\mathop{\rm ln}  \| X ( iT, ( i + 1) T) \|
 +
$$
  
 
(the lower central exponent); sometimes the lower central exponent is defined as
 
(the lower central exponent); sometimes the lower central exponent is defined as
  
<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/c/c021/c021170/c0211703.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {T \rightarrow + \infty } \
 +
\lim\limits _ {\overline{ {k \rightarrow + \infty }}\; } \
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c0211704.png" /> is the [[Cauchy operator|Cauchy operator]] of the system
+
\frac{- 1 }{kT }
  
<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/c/c021/c021170/c0211705.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\sum _ {i = 0 } ^ { {k }  - 1 }
 +
\mathop{\rm ln}  \| X ( iT, ( i + 1) T) \| .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c0211706.png" /> is a mapping
+
Here  $  X ( \theta , \tau ) $
 +
is the [[Cauchy operator|Cauchy operator]] 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/c/c021/c021170/c0211707.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\dot{x}  = A ( t) x,\ \
 +
x \in \mathbf R  ^ {n} ,
 +
$$
  
that is summable on every interval. The central exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c0211708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c0211709.png" /> may be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117010.png" />; the inequalities
+
where  $  A ( \cdot ) $
 +
is a mapping
  
<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/c/c021/c021170/c02117011.png" /></td> </tr></table>
+
$$
 +
\mathbf R  ^ {+}  \rightarrow \
 +
\mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} )
 +
$$
  
<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/c/c021/c021170/c02117012.png" /></td> </tr></table>
+
that is summable on every interval. The central exponents  $  \Omega ( A) $
 +
and  $  \omega ( A) $
 +
may be  $  \pm  \infty $;  
 +
the inequalities
 +
 
 +
$$
 +
\overline{\lim\limits}\; _ {t \rightarrow + \infty } \
 +
{
 +
\frac{1}{t}
 +
}
 +
\int\limits _ { 0 } ^ { t }
 +
\| A ( \tau ) \| \
 +
d \tau  \geq  \Omega ( A)  \geq  \
 +
\omega ( A) \geq
 +
$$
 +
 
 +
$$
 +
\geq  \
 +
- \lim\limits _ {\overline{ {t \rightarrow + \infty }}\; } \
 +
{
 +
\frac{1}{t}
 +
} \int\limits _ { 0 } ^ { t }  \| A ( \tau ) \|  d \tau
 +
$$
  
 
hold, which imply that if the system (1) satisfies the condition
 
hold, which imply that if the system (1) satisfies the condition
  
<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/c/c021/c021170/c02117013.png" /></td> </tr></table>
+
$$
 +
\overline{\lim\limits}\; _ {t \rightarrow + \infty } \
 +
{
 +
\frac{1}{t}
 +
} \int\limits _ { 0 } ^ { t }
 +
\| A ( \tau) \| \
 +
d \tau  < + \infty ,
 +
$$
  
then its central exponents are finite numbers. The central exponents are connected with the Lyapunov characteristic exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117014.png" /> and with the [[Singular exponents|singular exponents]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117015.png" /> by the inequalities
+
then its central exponents are finite numbers. The central exponents are connected with the Lyapunov characteristic exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]) $  \lambda _ {1} ( A) \dots \lambda _ {n} ( A) $
 +
and with the [[Singular exponents|singular exponents]] $  \Omega  ^ {0} ( A), \omega  ^ {0} ( A) $
 +
by 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/c/c021/c021170/c02117016.png" /></td> </tr></table>
+
$$
 +
\Omega  ^ {0} ( A)  \geq  \
 +
\Omega ( A)  \geq  \
 +
\lambda _ {1} ( A)  \geq  \dots \geq  \
 +
\lambda _ {n} ( A)  \geq  \
 +
\omega ( A)  \geq  \
 +
\omega  ^ {0} ( A).
 +
$$
  
For a system (1) with constant coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117017.png" /> the central exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117019.png" /> are equal, respectively, to the maximum and minimum of the real parts of the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117020.png" />. For a system (1) with periodic coefficients (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117021.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117022.png" /> and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117024.png" /> being the smallest period) the central exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117026.png" /> are equal, respectively, to the maximum and minimum of the logarithms of the moduli of the [[Multipliers|multipliers]] divided by the period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117027.png" />.
+
For a system (1) with constant coefficients $  ( A ( t) \equiv A) $
 +
the central exponents $  \Omega ( A) $
 +
and $  \omega ( A) $
 +
are equal, respectively, to the maximum and minimum of the real parts of the eigen values of $  A $.  
 +
For a system (1) with periodic coefficients ( $  A ( t + \theta ) = A ( t) $
 +
for all $  t \in \mathbf R $
 +
and some $  \theta > 0 $,  
 +
$  \theta $
 +
being the smallest period) the central exponents $  \Omega ( A) $
 +
and $  \omega ( A) $
 +
are equal, respectively, to the maximum and minimum of the logarithms of the moduli of the [[Multipliers|multipliers]] divided by the period $  \theta $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117028.png" /> is an almost-periodic mapping (see [[Linear system of differential equations with almost-periodic coefficients|Linear system of differential equations with almost-periodic coefficients]]), then the central exponents of (1) coincide with the singular exponents:
+
If $  A ( \cdot ) $
 +
is an almost-periodic mapping (see [[Linear system of differential equations with almost-periodic coefficients|Linear system of differential equations with almost-periodic coefficients]]), then the central exponents of (1) coincide with the singular exponents:
  
<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/c/c021/c021170/c02117029.png" /></td> </tr></table>
+
$$
 +
\Omega ( A)  = \
 +
\Omega  ^ {0} ( A),\ \
 +
\omega ( A)  = \
 +
\omega  ^ {0} ( A)
 +
$$
  
 
(Bylov's theorem).
 
(Bylov's theorem).
  
For every fixed system (1) the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117030.png" /> is sufficient for the existence of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117031.png" /> such that for every system
+
For every fixed system (1) the condition $  \Omega ( A) < 0 $
 +
is sufficient for the existence of a $  \delta > 0 $
 +
such that for every 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/c/c021/c021170/c02117032.png" /></td> </tr></table>
+
$$
 +
\dot{x}  = \
 +
A ( t) x + g ( x, t)
 +
$$
  
 
satisfying the conditions of the existence and uniqueness theorem for the solution of the Cauchy problem and the condition
 
satisfying the conditions of the existence and uniqueness theorem for the solution of the Cauchy problem and the condition
  
<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/c/c021/c021170/c02117033.png" /></td> </tr></table>
+
$$
 +
| g ( x, t) |  < \delta | x | ,
 +
$$
  
the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117034.png" /> is asymptotically stable (Vinograd's theorem). The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117035.png" /> in Vinograd's theorem is not only sufficient but also necessary. (The necessity remains valid when asymptotic stability is replaced by Lyapunov stability.)
+
the solution $  x = 0 $
 +
is asymptotically stable (Vinograd's theorem). The condition $  \Omega ( A) < 0 $
 +
in Vinograd's theorem is not only sufficient but also necessary. (The necessity remains valid when asymptotic stability is replaced by Lyapunov stability.)
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117036.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117037.png" />) on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117038.png" /> of the system (1) with bounded continuous coefficients (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117039.png" /> is continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117040.png" />), endowed with the metric
+
The function $  \Omega ( A) $(
 +
respectively, $  \omega ( A) $)  
 +
on the space $  M _ {n} $
 +
of the system (1) with bounded continuous coefficients (so that $  A ( \cdot ) $
 +
is continuous and $  \sup _ {t \in \mathbf R  ^ {+}  }  \| A ( t) \| < + \infty $),  
 +
endowed 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/c/c021/c021170/c02117041.png" /></td> </tr></table>
+
$$
 +
d ( A, B)  = \
 +
\sup _ {t \in \mathbf R  ^ {+} } \
 +
\| A ( t) - B ( t) \| ,
 +
$$
  
is upper (respectively lower) semi-continuous, but neither of these functions is continuous everywhere. For every system (1), in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117042.png" /> one can find another system
+
is upper (respectively lower) semi-continuous, but neither of these functions is continuous everywhere. For every system (1), in $  M _ {n} $
 +
one can find another 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/c/c021/c021170/c02117043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\dot{x}  = B _ {i} ( t) x,\ \
 +
i = 1, 2,
 +
$$
  
arbitrarily close to it (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117044.png" />) such that
+
arbitrarily close to it (in $  M _ {n} $)  
 +
such that
  
<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/c/c021/c021170/c02117045.png" /></td> </tr></table>
+
$$
 +
\lambda _ {1} ( B _ {1} )  = \
 +
\Omega ( A),\ \
 +
\lambda _ {n} ( B _ {2} )  = \
 +
\omega ( A),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117048.png" />, are the largest (highest) and the smallest (least) characteristic Lyapunov exponents of the system (2).
+
where $  \lambda _ {1} ( B _ {i} ) $
 +
and $  \lambda _ {n} ( B _ {i} ) $,
 +
$  i = 1, 2 $,  
 +
are the largest (highest) and the smallest (least) characteristic Lyapunov exponents of the system (2).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117049.png" /> is a uniformly-continuous mapping and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117050.png" />, then for almost-every mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117051.png" /> (in the sense of every normalized invariant measure of shift dynamical systems (cf. [[Shift dynamical system|Shift dynamical system]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117052.png" />, concentrated on the closure of the trajectory of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117053.png" />; the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117055.png" /> are regarded as points of the space of the shift dynamical system) the upper (lower) central exponent of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117056.png" /> is equal to the largest (smallest) characteristic Lyapunov exponent of this system:
+
If $  A ( \cdot ): \mathbf R \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $
 +
is a uniformly-continuous mapping and if $  \sup _ {t \in \mathbf R }  \| A ( t) \| < + \infty $,  
 +
then for almost-every mapping $  \widetilde{A}  $(
 +
in the sense of every normalized invariant measure of shift dynamical systems (cf. [[Shift dynamical system|Shift dynamical system]]), $  S = \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $,  
 +
concentrated on the closure of the trajectory of the point $  A $;  
 +
the mappings $  \widetilde{A}  $
 +
and $  A $
 +
are regarded as points of the space of the shift dynamical system) the upper (lower) central exponent of the system $  \dot{x} = \widetilde{A}  ( t) x $
 +
is equal to the largest (smallest) characteristic Lyapunov exponent of this 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/c/c021/c021170/c02117057.png" /></td> </tr></table>
+
$$
 +
\Omega ( \widetilde{A}  )  = \
 +
\lambda _ {1} ( \widetilde{A}  ),\ \
 +
\omega ( \widetilde{A}  )  = \
 +
\lambda _ {n} ( \widetilde{A}  ).
 +
$$
  
Suppose that a dynamical system on a smooth closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117058.png" /> is given by a smooth vector field. Then for almost-every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117059.png" /> (in the sense of every normalized invariant measure) the upper (lower) central exponent of the system of equations in variations along the trajectory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117060.png" /> coincides with its largest (smallest) characteristic Lyapunov exponent. Generic properties of the central exponent (from the point of view of the Baire categories) have been studied, see .
+
Suppose that a dynamical system on a smooth closed manifold $  V  ^ {n} $
 +
is given by a smooth vector field. Then for almost-every point $  x \in V  ^ {n} $(
 +
in the sense of every normalized invariant measure) the upper (lower) central exponent of the system of equations in variations along the trajectory of $  x $
 +
coincides with its largest (smallest) characteristic Lyapunov exponent. Generic properties of the central exponent (from the point of view of the Baire categories) have been studied, see .
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  N.A. Izobov,  "Linear systems of ordinary differential equations"  ''J. Soviet Math.'' , '''5''' :  1  (1974)  pp. 46–96  ''Itogi Nauk. Mat. Anal'' , '''12'''  pp. 71–146</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  V.M. Millionshchikov,  "Typical properties of conditional exponential stability II"  ''Differential equations'' , '''19''' :  9  pp. 1126–1132  ''Differentsial'nye Uravneniya'' , '''19''' :  9  (1983)  pp. 1503–1510</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  V.M. Millionshchikov,  "Typical properties of conditional exponential stability VI"  ''Differential equations'' , '''20''' :  6  pp. 707–715  ''Differentsial'nye Uravneniya'' , '''20''' :  6  (1984)</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top">  V.M. Millionshchikov,  "Typical properties of conditional exponential stability VII"  ''Differential equations'' , '''20''' :  8  pp. 1005–1013  ''Differentsial'nye Uravneniya'' , '''20''' :  8  (1984)  pp. 1366–1376</TD></TR></table>
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  N.A. Izobov,  "Linear systems of ordinary differential equations"  ''J. Soviet Math.'' , '''5''' :  1  (1974)  pp. 46–96  ''Itogi Nauk. Mat. Anal'' , '''12'''  pp. 71–146</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  V.M. Millionshchikov,  "Typical properties of conditional exponential stability II"  ''Differential equations'' , '''19''' :  9  pp. 1126–1132  ''Differentsial'nye Uravneniya'' , '''19''' :  9  (1983)  pp. 1503–1510</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  V.M. Millionshchikov,  "Typical properties of conditional exponential stability VI"  ''Differential equations'' , '''20''' :  6  pp. 707–715  ''Differentsial'nye Uravneniya'' , '''20''' :  6  (1984)</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top">  V.M. Millionshchikov,  "Typical properties of conditional exponential stability VII"  ''Differential equations'' , '''20''' :  8  pp. 1005–1013  ''Differentsial'nye Uravneniya'' , '''20''' :  8  (1984)  pp. 1366–1376</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>

Latest revision as of 16:43, 4 June 2020


of a linear system of ordinary differential equations

Quantities defined by the formulas

$$ \Omega ( A) = \ \lim\limits _ {T \rightarrow + \infty } \ \overline{\lim\limits}\; _ {k \rightarrow + \infty } \ { \frac{1}{kT} } \sum _ {i = 0 } ^ { {k } - 1 } \mathop{\rm ln} \| X (( i + 1) T, iT) \| $$

(the upper central exponent) and

$$ \omega ( A) = \ \lim\limits _ {T \rightarrow + \infty } \ \overline{\lim\limits}\; _ {k \rightarrow + \infty } \ \frac{- 1 }{kT } \sum _ {i = 0 } ^ { {k } - 1 } \mathop{\rm ln} \| X ( iT, ( i + 1) T) \| $$

(the lower central exponent); sometimes the lower central exponent is defined as

$$ \lim\limits _ {T \rightarrow + \infty } \ \lim\limits _ {\overline{ {k \rightarrow + \infty }}\; } \ \frac{- 1 }{kT } \sum _ {i = 0 } ^ { {k } - 1 } \mathop{\rm ln} \| X ( iT, ( i + 1) T) \| . $$

Here $ X ( \theta , \tau ) $ is the Cauchy operator 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 central exponents $ \Omega ( A) $ and $ \omega ( A) $ may be $ \pm \infty $; the inequalities

$$ \overline{\lim\limits}\; _ {t \rightarrow + \infty } \ { \frac{1}{t} } \int\limits _ { 0 } ^ { t } \| A ( \tau ) \| \ d \tau \geq \Omega ( A) \geq \ \omega ( A) \geq $$

$$ \geq \ - \lim\limits _ {\overline{ {t \rightarrow + \infty }}\; } \ { \frac{1}{t} } \int\limits _ { 0 } ^ { t } \| A ( \tau ) \| d \tau $$

hold, which imply that if the system (1) satisfies the condition

$$ \overline{\lim\limits}\; _ {t \rightarrow + \infty } \ { \frac{1}{t} } \int\limits _ { 0 } ^ { t } \| A ( \tau) \| \ d \tau < + \infty , $$

then its central exponents are finite numbers. The central exponents are connected with the Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent) $ \lambda _ {1} ( A) \dots \lambda _ {n} ( A) $ and with the singular exponents $ \Omega ^ {0} ( A), \omega ^ {0} ( A) $ by the inequalities

$$ \Omega ^ {0} ( A) \geq \ \Omega ( A) \geq \ \lambda _ {1} ( A) \geq \dots \geq \ \lambda _ {n} ( A) \geq \ \omega ( A) \geq \ \omega ^ {0} ( A). $$

For a system (1) with constant coefficients $ ( A ( t) \equiv A) $ the central exponents $ \Omega ( A) $ and $ \omega ( A) $ are equal, respectively, to the maximum and minimum of the real parts of the eigen values of $ A $. For a system (1) with periodic coefficients ( $ A ( t + \theta ) = A ( t) $ for all $ t \in \mathbf R $ and some $ \theta > 0 $, $ \theta $ being the smallest period) the central exponents $ \Omega ( A) $ and $ \omega ( A) $ are equal, respectively, to the maximum and minimum of the logarithms of the moduli of the multipliers divided by the period $ \theta $.

If $ A ( \cdot ) $ is an almost-periodic mapping (see Linear system of differential equations with almost-periodic coefficients), then the central exponents of (1) coincide with the singular exponents:

$$ \Omega ( A) = \ \Omega ^ {0} ( A),\ \ \omega ( A) = \ \omega ^ {0} ( A) $$

(Bylov's theorem).

For every fixed system (1) the condition $ \Omega ( A) < 0 $ is sufficient for the existence of a $ \delta > 0 $ such that for every system

$$ \dot{x} = \ A ( t) x + g ( x, t) $$

satisfying the conditions of the existence and uniqueness theorem for the solution of the Cauchy problem and the condition

$$ | g ( x, t) | < \delta | x | , $$

the solution $ x = 0 $ is asymptotically stable (Vinograd's theorem). The condition $ \Omega ( A) < 0 $ in Vinograd's theorem is not only sufficient but also necessary. (The necessity remains valid when asymptotic stability is replaced by Lyapunov stability.)

The function $ \Omega ( A) $( respectively, $ \omega ( A) $) on the space $ M _ {n} $ of the system (1) with bounded continuous coefficients (so that $ A ( \cdot ) $ is continuous and $ \sup _ {t \in \mathbf R ^ {+} } \| A ( t) \| < + \infty $), endowed with the metric

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

is upper (respectively lower) semi-continuous, but neither of these functions is continuous everywhere. For every system (1), in $ M _ {n} $ one can find another system

$$ \tag{2 } \dot{x} = B _ {i} ( t) x,\ \ i = 1, 2, $$

arbitrarily close to it (in $ M _ {n} $) such that

$$ \lambda _ {1} ( B _ {1} ) = \ \Omega ( A),\ \ \lambda _ {n} ( B _ {2} ) = \ \omega ( A), $$

where $ \lambda _ {1} ( B _ {i} ) $ and $ \lambda _ {n} ( B _ {i} ) $, $ i = 1, 2 $, are the largest (highest) and the smallest (least) characteristic Lyapunov exponents of the system (2).

If $ A ( \cdot ): \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $ is a uniformly-continuous mapping and if $ \sup _ {t \in \mathbf R } \| A ( t) \| < + \infty $, then for almost-every mapping $ \widetilde{A} $( in the sense of every normalized invariant measure of shift dynamical systems (cf. Shift dynamical system), $ S = \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $, concentrated on the closure of the trajectory of the point $ A $; the mappings $ \widetilde{A} $ and $ A $ are regarded as points of the space of the shift dynamical system) the upper (lower) central exponent of the system $ \dot{x} = \widetilde{A} ( t) x $ is equal to the largest (smallest) characteristic Lyapunov exponent of this system:

$$ \Omega ( \widetilde{A} ) = \ \lambda _ {1} ( \widetilde{A} ),\ \ \omega ( \widetilde{A} ) = \ \lambda _ {n} ( \widetilde{A} ). $$

Suppose that a dynamical system on a smooth closed manifold $ V ^ {n} $ is given by a smooth vector field. Then for almost-every point $ x \in V ^ {n} $( in the sense of every normalized invariant measure) the upper (lower) central exponent of the system of equations in variations along the trajectory of $ x $ coincides with its largest (smallest) characteristic Lyapunov exponent. Generic properties of the central exponent (from the point of view of the Baire categories) have been studied, see .

References

[1] 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)
[2] N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 : 1 (1974) pp. 46–96 Itogi Nauk. Mat. Anal , 12 pp. 71–146
[3a] V.M. Millionshchikov, "Typical properties of conditional exponential stability II" Differential equations , 19 : 9 pp. 1126–1132 Differentsial'nye Uravneniya , 19 : 9 (1983) pp. 1503–1510
[3b] V.M. Millionshchikov, "Typical properties of conditional exponential stability VI" Differential equations , 20 : 6 pp. 707–715 Differentsial'nye Uravneniya , 20 : 6 (1984)
[3c] V.M. Millionshchikov, "Typical properties of conditional exponential stability VII" Differential equations , 20 : 8 pp. 1005–1013 Differentsial'nye Uravneniya , 20 : 8 (1984) pp. 1366–1376

Comments

References

[a1] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
How to Cite This Entry:
Central exponents. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Central_exponents&oldid=46294
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article