View source for Stability of characteristic exponents

<!--
s0870201.png
$#A+1 = 52 n = 0
$#C+1 = 52 : ~/encyclopedia/old_files/data/S087/S.0807020 Stability of characteristic exponents
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A property of the (Lyapunov) characteristic exponents (cf. [[Lyapunov characteristic exponent|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. [[#References|[2]]], [[#References|[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|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|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:

$$ \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|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|variational equations]] associated with the trajectory of the point  $  x $
are stochastically stable.

====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>