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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

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

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. [[#References|[2]]], [[#References|[3]]]). For example, the characteristic exponents of the system

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

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 ( \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]]).

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:

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

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|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 \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|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|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|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>