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Difference between revisions of "Stability of characteristic exponents"

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A property of the (Lyapunov) characteristic exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]) of a linear system of ordinary differential equations
 
A property of the (Lyapunov) characteristic exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]) of a linear system of ordinary differential equations
  
<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/s087/s087020/s0870201.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/s087/s087020/s0870202.png" /> is a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s0870203.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s0870204.png" />), satisfying the condition
+
where $  A ( \cdot ) $
 +
is a continuous mapping $  \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} ) $),  
 +
satisfying 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/s/s087/s087020/s0870205.png" /></td> </tr></table>
+
$$
 +
\sup _ { t \in \mathbf R  ^ {+} }  \| A ( t) \|  < + \infty .
 +
$$
  
 
One says that the characteristic exponents of the system (1) are stable if each of the functions
 
One says that the characteristic exponents of the system (1) are stable if each of the functions
  
<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/s087/s087020/s0870206.png" /></td> </tr></table>
+
$$
 +
\lambda _ {i} ( \cdot ): M _ {n}  \rightarrow  \mathbf R ,\ \
 +
i = 1 \dots n,
 +
$$
  
is continuous at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s0870207.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s0870208.png" /> are the characteristic exponents of the system (1) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s0870209.png" /> is the set of all systems (1), equipped with the structure of a metric space given by the distance
+
is continuous at the point $  A $.  
 +
Here $  \lambda _ {1} ( A) \geq  \dots \geq  \lambda _ {n} ( A) $
 +
are the characteristic exponents of the system (1) and $  M _ {n} $
 +
is the set of all systems (1), equipped with the structure of a metric space given by the distance
  
<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/s087/s087020/s08702010.png" /></td> </tr></table>
+
$$
 +
d ( A, B)  = \
 +
\sup _ { t \in \mathbf R  ^ {+} }  \| A ( t) - B ( t) \|
 +
$$
  
(for convenience the system (1) is identified with the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702011.png" />; moreover, instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702012.png" /> one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702013.png" />).
+
(for convenience the system (1) is identified with the mapping $  A ( \cdot ) $;  
 +
moreover, instead of $  A ( \cdot ) $
 +
one writes $  A $).
  
 
Systems (1) with unstable exponents have been found (cf. [[#References|[2]]], [[#References|[3]]]). For example, the characteristic exponents of the system
 
Systems (1) with unstable exponents have been found (cf. [[#References|[2]]], [[#References|[3]]]). For example, the characteristic exponents 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/s087/s087020/s08702014.png" /></td> </tr></table>
+
$$
 +
\dot{v}  = \
 +
( \sin  \mathop{\rm ln} ( 1 + t) + \cos  \mathop{\rm ln} ( 1 + t)) v + \delta u
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702015.png" /> are unstable, since for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702016.png" /> the largest characteristic exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702017.png" /> is 1, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702020.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702021.png" />. For the stability of the characteristic exponents it is sufficient that the [[Integral separation condition|integral separation condition]] should be fulfilled (Perron's theorem). The set of systems (1) satisfying this condition coincides with the interior (in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702022.png" />) of the set of all systems (1) with stable characteristic exponents.
+
for $  \delta = 0 $
 +
are unstable, since for $  \delta = 0 $
 +
the largest characteristic exponent $  \lambda _ {1} $
 +
is 1, and for $  \delta \neq 0 $,  
 +
$  \lambda _ {1} > 1 $
 +
and $  \lambda _ {1} $
 +
does not depend on $  \delta \neq 0 $.  
 +
For the stability of the characteristic exponents it is sufficient that the [[Integral separation condition|integral separation condition]] should be fulfilled (Perron's theorem). The set of systems (1) satisfying this condition coincides with the interior (in the space $  M _ {n} $)  
 +
of the set of all systems (1) with stable characteristic exponents.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702023.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702026.png" /> (for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702027.png" />) (i.e. the system (1) has constant or periodic coefficients), then the characteristic exponents of the system (1) are stable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702028.png" /> is an almost-periodic mapping (cf. [[Linear system of differential equations with almost-periodic coefficients|Linear system of differential equations with almost-periodic coefficients]]), then for the stability of the characteristic exponents of the system (1) it is necessary and sufficient that the system (1) be almost reducible (cf. also [[Reducible linear system|Reducible linear system]]).
+
If $  A ( t) \equiv A ( 0) $
 +
for all $  t \in \mathbf R  ^ {n} $
 +
or $  A ( t + T) = A ( t) $
 +
for all $  t \in \mathbf R  ^ {n} $(
 +
for a certain $  T > 0 $)  
 +
(i.e. the system (1) has constant or periodic coefficients), then the characteristic exponents of the system (1) are stable. If $  A ( \cdot ) $
 +
is an almost-periodic mapping (cf. [[Linear system of differential equations with almost-periodic coefficients|Linear system of differential equations with almost-periodic coefficients]]), then for the stability of the characteristic exponents of the system (1) it is necessary and sufficient that the system (1) be almost reducible (cf. also [[Reducible linear system|Reducible linear system]]).
  
 
For the characteristic exponents of the system (1) to be stable it is sufficient that there is a [[Lyapunov transformation|Lyapunov transformation]] reducing the system (1) to block-diagonal form:
 
For the characteristic exponents of the system (1) to be stable it is sufficient that there is a [[Lyapunov transformation|Lyapunov transformation]] reducing the system (1) to block-diagonal form:
  
<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/s087/s087020/s08702029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\left .
  
such that: a) the blocks are integrally separable, i.e. numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702031.png" /> can be found such that
+
such that: a) the blocks are integrally separable, i.e. numbers $  a > 0 $,
 +
$  d > 0 $
 +
can be found 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/s/s087/s087020/s08702032.png" /></td> </tr></table>
+
$$
 +
\| Y _ {i} ( \tau , \theta ) \|  ^ {-} 1  \geq  \
 +
d (  \mathop{\rm exp} [ a \cdot ( \theta - \tau )]) \
 +
\| Y _ {i + 1 }  ( \theta , \tau ) \|
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702034.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702035.png" /> is the [[Cauchy operator|Cauchy operator]] for the system (2)); and b) the upper and lower [[Central exponents|central exponents]] of the system (2) are equal to each other:
+
for all $  \theta \geq  \tau \geq  0 $,  
 +
$  i = 1 \dots m - 1 $(
 +
here $  Y _ {j} ( \theta , \tau ) $
 +
is the [[Cauchy operator|Cauchy operator]] for the system (2)); and b) the upper and lower [[Central exponents|central exponents]] of the system (2) are equal to each other:
  
<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/s087/s087020/s08702036.png" /></td> </tr></table>
+
$$
 +
\Omega ( B _ {i} )  = \omega ( B _ {i} ) \ \
 +
\textrm{ for }  \textrm{ each }  i = 1 \dots m.
 +
$$
  
 
The conditions of this theorem are also necessary for the stability of the characteristic exponents of the system (1) (cf. [[#References|[6]]]). Systems with unstable characteristic exponents may possess the property of stochastic stability of the characteristic exponents.
 
The conditions of this theorem are also necessary for the stability of the characteristic exponents of the system (1) (cf. [[#References|[6]]]). Systems with unstable characteristic exponents may possess the property of stochastic stability of the characteristic exponents.
  
The characteristic exponents of the system (1) are called stochastically stable (or almost-certainly stable) if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702037.png" /> the characteristic exponents of the system
+
The characteristic exponents of the system (1) are called stochastically stable (or almost-certainly stable) if for $  \sigma \rightarrow 0 $
 +
the characteristic exponents 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/s087/s087020/s08702038.png" /></td> </tr></table>
+
$$
 +
\dot{y}  = A ( t) y + \sigma  ^ {2} C ( t, \omega ) y
 +
$$
  
tend with probability 1 to the characteristic exponents of the system (1); here the elements of the matrix giving the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702039.png" /> (in a certain basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702040.png" /> which is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702041.png" />) are independent non-null [[White noise|white noise]].
+
tend with probability 1 to the characteristic exponents of the system (1); here the elements of the matrix giving the linear operator $  C ( t, \omega ): \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $(
 +
in a certain basis of $  \mathbf R  ^ {n} $
 +
which is independent of $  ( t, \omega ) $)  
 +
are independent non-null [[White noise|white noise]].
  
If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702042.png" /> is uniformly continuous and if
+
If the mapping $  A ( \cdot ): \mathbf R \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $
 +
is uniformly continuous and if
  
<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/s087/s087020/s08702043.png" /></td> </tr></table>
+
$$
 +
\sup _ {t \in \mathbf R }  \| A ( t) \|  < + \infty ,
 +
$$
  
then for almost-every mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702044.png" />, where
+
then for almost-every mapping $  \widetilde{A}  ( \cdot ) $,  
 +
where
  
<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/s087/s087020/s08702045.png" /></td> </tr></table>
+
$$
 +
\widetilde{A}  ( t)  = \lim\limits _ {k \rightarrow \infty }  A ( t _ {k} + t),
 +
$$
  
the characteristic exponents of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702046.png" /> are stochastically stable (for the [[Shift dynamical system|shift dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702047.png" /> one considers a normalized invariant measure, concentrated on the closure of the trajectory of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702048.png" />; by  "almost-every A"  one means almost-every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702049.png" /> in the sense of each such measure).
+
the characteristic exponents of the system $  \dot{x} = A ( t) x $
 +
are stochastically stable (for the [[Shift dynamical system|shift dynamical system]] $  ( S = \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} )) $
 +
one considers a normalized invariant measure, concentrated on the closure of the trajectory of the point $  A ( \cdot ) $;  
 +
by  "almost-every A"  one means almost-every $  A ( \cdot ) $
 +
in the sense of each such measure).
  
Let a dynamical system on a smooth closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702050.png" /> be given by a smooth vector field. Then for almost-every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702051.png" /> (in the sense of each normalized invariant measure) the characteristic exponents of the system of [[Variational equations|variational equations]] associated with the trajectory of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087020/s08702052.png" /> are stochastically stable.
+
Let a dynamical system on a smooth closed manifold $  V  ^ {n} $
 +
be given by a smooth vector field. Then for almost-every point $  x \in V  ^ {n} $(
 +
in the sense of each normalized invariant measure) the characteristic exponents of the system of [[Variational equations|variational equations]] associated with the trajectory of the point $  x $
 +
are stochastically stable.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Perron,  "Die Ordnungszahlen der Differentialgleichungssysteme"  ''Math. Z.'' , '''31'''  (1930)  pp. 748–766</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Perron,  "Ueber lineare Differentialgleichungen, bei denen die unabhängig Variable reel ist I"  ''J. Reine Angew. Math.'' , '''142'''  (1913)  pp. 254–270</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</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">[6]</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></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Perron,  "Die Ordnungszahlen der Differentialgleichungssysteme"  ''Math. Z.'' , '''31'''  (1930)  pp. 748–766</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Perron,  "Ueber lineare Differentialgleichungen, bei denen die unabhängig Variable reel ist I"  ''J. Reine Angew. Math.'' , '''142'''  (1913)  pp. 254–270</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</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">[6]</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></table>

Revision as of 08:22, 6 June 2020


A property of the (Lyapunov) characteristic exponents (cf. Lyapunov characteristic exponent) of a linear system of ordinary differential equations

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

where $ A ( \cdot ) $ is a continuous mapping $ \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} ) $), satisfying the condition

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

One says that the characteristic exponents of the system (1) are stable if each of the functions

$$ \lambda _ {i} ( \cdot ): M _ {n} \rightarrow \mathbf R ,\ \ i = 1 \dots n, $$

is continuous at the point $ A $. Here $ \lambda _ {1} ( A) \geq \dots \geq \lambda _ {n} ( A) $ are the characteristic exponents of the system (1) and $ M _ {n} $ is the set of all systems (1), equipped with the structure of a metric space given by the distance

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

(for convenience the system (1) is identified with the mapping $ A ( \cdot ) $; moreover, instead of $ A ( \cdot ) $ one writes $ A $).

Systems (1) with unstable exponents have been found (cf. [2], [3]). For example, the characteristic exponents of the system

$$ \dot{v} = \ ( \sin \mathop{\rm ln} ( 1 + t) + \cos \mathop{\rm ln} ( 1 + t)) v + \delta u $$

for $ \delta = 0 $ are unstable, since for $ \delta = 0 $ the largest characteristic exponent $ \lambda _ {1} $ is 1, and for $ \delta \neq 0 $, $ \lambda _ {1} > 1 $ and $ \lambda _ {1} $ does not depend on $ \delta \neq 0 $. For the stability of the characteristic exponents it is sufficient that the integral separation condition should be fulfilled (Perron's theorem). The set of systems (1) satisfying this condition coincides with the interior (in the space $ M _ {n} $) of the set of all systems (1) with stable characteristic exponents.

If $ A ( t) \equiv A ( 0) $ for all $ t \in \mathbf R ^ {n} $ or $ A ( t + T) = A ( t) $ for all $ t \in \mathbf R ^ {n} $( for a certain $ T > 0 $) (i.e. the system (1) has constant or periodic coefficients), then the characteristic exponents of the system (1) are stable. If $ A ( \cdot ) $ is an almost-periodic mapping (cf. Linear system of differential equations with almost-periodic coefficients), then for the stability of the characteristic exponents of the system (1) it is necessary and sufficient that the system (1) be almost reducible (cf. also Reducible linear system).

For the characteristic exponents of the system (1) to be stable it is sufficient that there is a Lyapunov transformation reducing the system (1) to block-diagonal form:

$$ \tag{2 } \left . such that: a) the blocks are integrally separable, i.e. numbers $ a > 0 $, $ d > 0 $ can be found such that $$ \| Y _ {i} ( \tau , \theta ) \| ^ {-} 1 \geq \ d ( \mathop{\rm exp} [ a \cdot ( \theta - \tau )]) \ \| Y _ {i + 1 } ( \theta , \tau ) \| $$ for all $ \theta \geq \tau \geq 0 $, $ i = 1 \dots m - 1 $( here $ Y _ {j} ( \theta , \tau ) $ is the [[Cauchy operator|Cauchy operator]] for the system (2)); and b) the upper and lower [[Central exponents|central exponents]] of the system (2) are equal to each other: $$ \Omega ( B _ {i} ) = \omega ( B _ {i} ) \ \ \textrm{ for } \textrm{ each } i = 1 \dots m. $$ The conditions of this theorem are also necessary for the stability of the characteristic exponents of the system (1) (cf. [[#References|[6]]]). Systems with unstable characteristic exponents may possess the property of stochastic stability of the characteristic exponents. The characteristic exponents of the system (1) are called stochastically stable (or almost-certainly stable) if for $ \sigma \rightarrow 0 $ the characteristic exponents of the system $$ \dot{y} = A ( t) y + \sigma ^ {2} C ( t, \omega ) y $$ tend with probability 1 to the characteristic exponents of the system (1); here the elements of the matrix giving the linear operator $ C ( t, \omega ): \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $( in a certain basis of $ \mathbf R ^ {n} $ which is independent of $ ( t, \omega ) $) are independent non-null [[White noise|white noise]]. If the mapping $ A ( \cdot ): \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $ is uniformly continuous and if $$ \sup _ {t \in \mathbf R } \| A ( t) \| < + \infty , $$ then for almost-every mapping $ \widetilde{A} ( \cdot ) $, where $$ \widetilde{A} ( t) = \lim\limits _ {k \rightarrow \infty } A ( t _ {k} + t), $$

the characteristic exponents of the system $ \dot{x} = A ( t) x $ are stochastically stable (for the shift dynamical system $ ( S = \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} )) $ one considers a normalized invariant measure, concentrated on the closure of the trajectory of the point $ A ( \cdot ) $; by "almost-every A" one means almost-every $ A ( \cdot ) $ in the sense of each such measure).

Let a dynamical system on a smooth closed manifold $ V ^ {n} $ be given by a smooth vector field. Then for almost-every point $ x \in V ^ {n} $( in the sense of each normalized invariant measure) the characteristic exponents of the system of variational equations associated with the trajectory of the point $ x $ are stochastically stable.

References

[1] A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[2] O. Perron, "Die Ordnungszahlen der Differentialgleichungssysteme" Math. Z. , 31 (1930) pp. 748–766
[3] O. Perron, "Ueber lineare Differentialgleichungen, bei denen die unabhängig Variable reel ist I" J. Reine Angew. Math. , 142 (1913) pp. 254–270
[4] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[5] 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)
[6] 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
How to Cite This Entry:
Stability of characteristic exponents. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stability_of_characteristic_exponents&oldid=18227
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article