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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
  
$$ \tag{1 }
+
$$ \tag{1} \dot{x}  =  A (t) x,\ x \in \mathbf R ^ {n} , $$
\dot{x}  =  A ( t) x,\ \
 
x \in \mathbf R ^ {n} ,
 
$$
 
  
where  $  A ( \cdot ) $
+
where  $  A ( \mathop \cdot \limits ) $
is a continuous mapping  $  \mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $(
+
is a continuous mapping  $  \mathbf R ^ {+} \mathop \rightarrow \limits {\mathop{\rm Hom}\nolimits} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $  
or  $  \mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf C ^ {n} , \mathbf C ^ {n} ) $),  
+
(or  $  \mathbf R ^ {+} \mathop \rightarrow \limits {\mathop{\rm Hom}\nolimits} ( \mathbf C ^ {n} , \mathbf C ^ {n} ) $),  
 
satisfying the condition
 
satisfying the condition
  
$$  
+
$$ \mathop{\rm sup} _ { t \in \mathbf R ^ {+}}  \| A (t) \|  <  + \inf . $$
\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
  
$$  
+
$$ \lambda _ {i} ( \mathop \cdot \limits ):  M _ {n}  \mathop \rightarrow \limits \mathbf R ,\ i = 1 \dots n, $$
\lambda _ {i} ( \cdot ):  M _ {n}  \rightarrow  \mathbf R ,\ \
 
i = 1 \dots n,
 
$$
 
  
 
is continuous at the point  $  A $.  
 
is continuous at the point  $  A $.  
Here  $  \lambda _ {1} ( A) \geq  \dots \geq  \lambda _ {n} ( A) $
+
Here  $  \lambda _ {1} (A) \geq  \dots \geq  \lambda _ {n} (A) $
 
are the characteristic exponents of the system (1) and  $  M _ {n} $
 
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
 
is the set of all systems (1), equipped with the structure of a metric space given by the distance
  
$$  
+
$$ d (A, B)  = \mathop{\rm sup} _ { t \in \mathbf R ^ {+}}  \| A (t) - B (t) \| $$
d ( A, B)  = \  
 
\sup _ { t \in \mathbf R ^ {+} }  \| A ( t) - B ( t) \|
 
$$
 
  
(for convenience the system (1) is identified with the mapping  $  A ( \cdot ) $;  
+
(for convenience the system (1) is identified with the mapping  $  A ( \mathop \cdot \limits ) $;  
moreover, instead of  $  A ( \cdot ) $
+
moreover, instead of  $  A ( \mathop \cdot \limits ) $
 
one writes  $  A $).
 
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
  
$$  
+
$$ \dot{v}  = ( \mathop{\rm sin\mathop{\rm ln} (1 + t) + \mathop{\rm cos\mathop{\rm ln} (1 + t)) v + \delta u $$
\dot{v}  = \  
 
( \sin   \mathop{\rm ln} ( 1 + t) + \cos   \mathop{\rm ln} ( 1 + t)) v + \delta u
 
$$
 
  
 
for  $  \delta = 0 $
 
for  $  \delta = 0 $
Line 65: Line 51:
 
of the set of all systems (1) with stable characteristic exponents.
 
of the set of all systems (1) with stable characteristic exponents.
  
If  $  A ( t) \equiv A ( 0) $
+
If  $  A (t) \equiv A (0) $
for all  $  t \in \mathbf R ^ {n} $
+
for all  $  t \in \mathbf R ^ {n} $
or  $  A ( t + T) = A ( t) $
+
or  $  A (t + T) = A (t) $
for all  $  t \in \mathbf R ^ {n} $(
+
for all  $  t \in \mathbf R ^ {n} $  
for a certain  $  T > 0 $)  
+
(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 ) $
+
(i.e. the system (1) has constant or periodic coefficients), then the characteristic exponents of the system (1) are stable. If  $  A ( \mathop \cdot \limits ) $
 
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]]).
 
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:
  
$$ \tag{2 }
+
$$ \tag{2} \left . \begin{array}{c} \dot{y} _ {i}  =  B _ {i} (t) y _ {i} ,  \\ y _ {i}  \in  \mathbf R ^ {k _ {i}} ,\ i = 1 \dots m,\ \sum _ {i = 1} ^ {m} k _ {i} = n ,  \\ \end{array} \right \} $$
\left .
 
 
 
\begin{array}{c}
 
\dot{y} _ {i}  =  B _ {i} ( t) y _ {i} ,  \\
 
y _ {i}  \in  \mathbf R ^ {k _ {i} } ,\ \
 
i = 1 \dots m,\ \
 
\sum _ {i = 1 } ^ { m }  
 
k _ {i} = n ,  \\
 
\end{array}
 
 
 
\right \}
 
$$
 
  
 
such that: a) the blocks are integrally separable, i.e. numbers  $  a > 0 $,  
 
such that: a) the blocks are integrally separable, i.e. numbers  $  a > 0 $,  
Line 93: Line 67:
 
can be found such that
 
can be found such that
  
$$  
+
$$ \| Y _ {i} ( \tau , \theta ) \| ^ {-1}  \geq   d ( \mathop{\rm exp} [a \mathop \cdot \limits ( \theta - \tau )]) \| Y _ {i + 1} ( \theta , \tau ) \| $$
\| 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 $,  
 
for all  $  \theta \geq  \tau \geq  0 $,  
$  i = 1 \dots m - 1 $(
+
$  i = 1 \dots m - 1 $  
here  $  Y _ {j} ( \theta , \tau ) $
+
(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:
 
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. $$
\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  $  \sigma \rightarrow 0 $
+
The characteristic exponents of the system (1) are called stochastically stable (or almost-certainly stable) if for  $  \sigma \mathop \rightarrow \limits 0 $
 
the characteristic exponents of the system
 
the characteristic exponents of the system
  
$$  
+
$$ \dot{y}  =  A (t) y + \sigma ^ {2} C (t, \omega ) y $$
\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} $(
+
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} \mathop \rightarrow \limits \mathbf R ^ {n} $  
in a certain basis of  $  \mathbf R ^ {n} $
+
(in a certain basis of  $  \mathbf R ^ {n} $
which is independent of  $  ( t, \omega ) $)  
+
which is independent of  $  (t, \omega ) $)  
 
are independent non-null [[White noise|white noise]].
 
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} ) $
+
If the mapping  $  A ( \mathop \cdot \limits ):  \mathbf R \mathop \rightarrow \limits {\mathop{\rm Hom}\nolimits} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $
 
is uniformly continuous and if
 
is uniformly continuous and if
  
$$  
+
$$ \mathop{\rm sup} _ {t \in \mathbf R}  \| A (t) \|  <  + \inf , $$
\sup _ {t \in \mathbf R }  \| A ( t) \|  <  + \infty ,
 
$$
 
  
then for almost-every mapping  $  \widetilde{A}  ( \cdot ) $,  
+
then for almost-every mapping  $  \widetilde{A}  ( \mathop \cdot \limits ) $,  
 
where
 
where
  
$$  
+
$$ \widetilde{A}  (t)  =  \mathop{\rm lim} _ {k \mathop \rightarrow \limits \inf}  A (t _ {k} + t), $$
\widetilde{A}  ( t)  =  \lim\limits _ {k \rightarrow \infty }  A ( t _ {k} + t),
 
$$
 
  
the characteristic exponents of the system  $  \dot{x} = A ( t) x $
+
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} )) $
+
are stochastically stable (for the [[Shift dynamical system|shift dynamical system]]  $  (S = {\mathop{\rm Hom}\nolimits} ( \mathbf R ^ {n} , \mathbf R ^ {n} )) $
one considers a normalized invariant measure, concentrated on the closure of the trajectory of the point  $  A ( \cdot ) $;  
+
one considers a normalized invariant measure, concentrated on the closure of the trajectory of the point  $  A ( \mathop \cdot \limits ) $;  
by  "almost-every A"  one means almost-every  $  A ( \cdot ) $
+
by  "almost-every A"  one means almost-every  $  A ( \mathop \cdot \limits ) $
 
in the sense of each such measure).
 
in the sense of each such measure).
  
Let a dynamical system on a smooth closed manifold  $  V ^ {n} $
+
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} $(
+
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 $
+
(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.
 
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>

Latest revision as of 11:27, 21 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 ( \mathop \cdot \limits ) $ is a continuous mapping $ \mathbf R ^ {+} \mathop \rightarrow \limits {\mathop{\rm Hom}\nolimits} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $ (or $ \mathbf R ^ {+} \mathop \rightarrow \limits {\mathop{\rm Hom}\nolimits} ( \mathbf C ^ {n} , \mathbf C ^ {n} ) $), satisfying the condition

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

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

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

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

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

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

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

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

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 ( \mathop \cdot \limits ) $ 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:

$$ \tag{2} \left . \begin{array}{c} \dot{y} _ {i} = B _ {i} (t) y _ {i} , \\ y _ {i} \in \mathbf R ^ {k _ {i}} ,\ i = 1 \dots m,\ \sum _ {i = 1} ^ {m} k _ {i} = n , \\ \end{array} \right \} $$

such that: a) the blocks are integrally separable, i.e. numbers $ a > 0 $, $ d > 0 $ can be found such that

$$ \| Y _ {i} ( \tau , \theta ) \| ^ {-1} \geq d ( \mathop{\rm exp} [a \mathop \cdot \limits ( \theta - \tau )]) \| Y _ {i + 1} ( \theta , \tau ) \| $$

for all $ \theta \geq \tau \geq 0 $, $ i = 1 \dots m - 1 $ (here $ Y _ {j} ( \theta , \tau ) $ is the Cauchy operator for the system (2)); and b) the upper and lower 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. [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 \mathop \rightarrow \limits 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} \mathop \rightarrow \limits \mathbf R ^ {n} $ (in a certain basis of $ \mathbf R ^ {n} $ which is independent of $ (t, \omega ) $) are independent non-null white noise.

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

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

then for almost-every mapping $ \widetilde{A} ( \mathop \cdot \limits ) $, where

$$ \widetilde{A} (t) = \mathop{\rm lim} _ {k \mathop \rightarrow \limits \inf} A (t _ {k} + t), $$

the characteristic exponents of the system $ \dot{x} = A (t) x $ are stochastically stable (for the shift dynamical system $ (S = {\mathop{\rm Hom}\nolimits} ( \mathbf R ^ {n} , \mathbf R ^ {n} )) $ one considers a normalized invariant measure, concentrated on the closure of the trajectory of the point $ A ( \mathop \cdot \limits ) $; by "almost-every A" one means almost-every $ A ( \mathop \cdot \limits ) $ 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=49598
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article