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''of a solution of a linear system''
 
''of a solution of a linear system''
  
The limes superior
+
The [[limes superior]]
 +
 
 +
$$
 +
\lambda _ {x ( t) }  = \
 +
{\lim\limits _ {t \rightarrow + \infty } } bar \
  
<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/l/l061/l061130/l0611301.png" /></td> </tr></table>
+
\frac{1}{t}
 +
\
 +
\mathop{\rm ln}  | x ( t) | ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l0611302.png" /> is a solution of the linear system of ordinary differential equations
+
where $  x ( t) \neq 0 $
 +
is a solution of the 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/l/l061/l061130/l0611303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\dot{x}  = A ( t) x ;
 +
$$
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l0611304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l0611305.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l0611306.png" />, summable on every interval. In coordinates,
+
here $  x \in \mathbf R  ^ {n} $
 +
and $  A ( \cdot ) $
 +
is a mapping $  \mathbf R \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $,
 +
summable on every interval. In coordinates,
  
<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/l/l061/l061130/l0611307.png" /></td> </tr></table>
+
$$
 +
x ( t)  = ( x  ^ {1} ( t) \dots x  ^ {n} ( t) ) ,
 +
$$
  
<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/l/l061/l061130/l0611308.png" /></td> </tr></table>
+
$$
 +
\dot{x}  ^ {i}  = \sum _ { j= } 1 ^ { n }  a _ {j}  ^ {i} ( t) x  ^ {j} ,\  i = 1 \dots n ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l0611309.png" /> are functions summable on every interval and
+
where $  a _ {j}  ^ {i} ( t) $
 +
are functions summable on every interval 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/l/l061/l061130/l06113010.png" /></td> </tr></table>
+
$$
 +
| x |  = \sqrt {\sum _ { i= } 1 ^ { n }  | x  ^ {i} |  ^ {2} }
 +
$$
  
(or any other equivalent norm; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113011.png" /> does not depend on the choice of the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113012.png" /> or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113013.png" />).
+
(or any other equivalent norm; $  \lambda _ {x ( t) }  $
 +
does not depend on the choice of the norm in $  \mathbf R  ^ {n} $
 +
or in $  \mathbf C  ^ {n} $).
  
 
Lyapunov's theorem. Suppose that
 
Lyapunov's theorem. Suppose 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/l/l061/l061130/l06113014.png" /></td> </tr></table>
+
$$
 +
{\lim\limits _ {t \rightarrow + \infty } } bar \
 +
 
 +
\frac{1}{t}
 +
\int\limits _ { 0 } ^ { t }
 +
\| A ( \tau ) \|  d \tau
 +
< + \infty ;
 +
$$
  
 
equivalently:
 
equivalently:
  
<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/l/l061/l061130/l06113015.png" /></td> </tr></table>
+
$$
 +
{\lim\limits _ {t \rightarrow + \infty } } bar \
 +
 
 +
\frac{1}{t}
 +
\int\limits _ { 0 } ^ { t }
 +
| a _ {j}  ^ {i} ( \tau ) |  d \tau  < + \infty ,\ \
 +
i , j = 1 \dots n .
 +
$$
  
Then for any solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113016.png" /> of the system (1) the Lyapunov characteristic exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113017.png" /> is a real number (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113018.png" />). The following assertions hold for the Lyapunov characteristic exponents of non-zero solutions of (1):
+
Then for any solution $  x ( t) \neq 0 $
 +
of the system (1) the Lyapunov characteristic exponent $  \lambda _ {x ( t) }  $
 +
is a real number (that is, $  \neq \pm  \infty $).  
 +
The following assertions hold for the Lyapunov characteristic exponents of non-zero solutions of (1):
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113020.png" />;
+
1) $  \lambda _ {\alpha x ( t) }  = \lambda _ {x ( t) }  $,  
 +
$  \alpha \neq 0 $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113021.png" />;
+
2) $  \lambda _ {x _ {1}  ( t) + x _ {2} ( t) } \leq  \max ( \lambda _ {x _ {1}  ( t) } , \lambda _ {x _ {2}  ( t) } ) $;
  
3) there exists a set of linearly independent solutions of (1), denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113022.png" />, such that for any other <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113023.png" /> linearly independent solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113025.png" />, of (1), numbered in decreasing order of the Lyapunov characteristic exponents, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113027.png" />, the following inequalities hold:
+
3) there exists a set of linearly independent solutions of (1), denoted by $  \{ x _ {i} ( t) \} _ {i=} 1  ^ {n} $,  
 +
such that for any other $  n $
 +
linearly independent solutions $  \widehat{x}  _ {i} ( t) $,  
 +
$  i = 1 \dots n $,  
 +
of (1), numbered in decreasing order of the Lyapunov characteristic exponents, that is, $  \lambda _ {\widehat{x}  _ {i}  ( t) } \geq  \lambda _ {\widehat{x}  _ {j}  ( t) } $
 +
for $  i \leq  j $,  
 +
the following inequalities hold:
  
<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/l/l061/l061130/l06113028.png" /></td> </tr></table>
+
$$
 +
\lambda _ {\widehat{x}  _ {i}  ( t) }  \geq  \
 +
\lambda _ {x _ {i}  ( t) } ,\ \
 +
i = 1 \dots n .
 +
$$
  
A [[Fundamental system of solutions|fundamental system of solutions]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113029.png" /> with this property is called normal. Such a normal system has the properties:
+
A [[Fundamental system of solutions|fundamental system of solutions]] $  \{ x _ {i} ( t) \} _ {i=} 1  ^ {n} $
 +
with this property is called normal. Such a normal system has the properties:
  
a) the family of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113031.png" />, does not depend on the choice of the normal fundamental system;
+
a) the family of numbers $  \lambda _ {i} ( A) = \lambda _ {x _ {i}  ( t) } $,  
 +
$  i = 1 \dots n $,  
 +
does not depend on the choice of the normal fundamental system;
  
b) for any solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113032.png" /> of (1) its Lyapunov characteristic exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113033.png" /> is equal to some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113034.png" />;
+
b) for any solution $  x ( t) \neq 0 $
 +
of (1) its Lyapunov characteristic exponent $  \lambda _ {x ( t) }  $
 +
is equal to some $  \lambda _ {i} ( A) $;
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113036.png" />.
+
c) $  \lambda _ {i} ( A) \geq  \lambda _ {j} ( A) $,  
 +
$  i \leq  j $.
  
The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113037.png" /> are called the Lyapunov characteristic exponents of the system (1); the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113038.png" /> is often called the leading Lyapunov characteristic exponent of (1).
+
The numbers $  \lambda _ {1} ( A) \geq  \dots \geq  \lambda _ {n} ( A) $
 +
are called the Lyapunov characteristic exponents of the system (1); the number $  \lambda _ {1} ( A) $
 +
is often called the leading Lyapunov characteristic exponent of (1).
  
 
The set of all Lyapunov characteristic exponents of non-zero solutions of (1) is called the spectrum.
 
The set of all Lyapunov characteristic exponents of non-zero solutions of (1) is called the spectrum.
Line 53: Line 123:
 
===Special case.===
 
===Special case.===
  
 +
1) A system with constant coefficients (that is,  $  A ( t) \equiv A ( 0) $).
 +
In this case the  $  \lambda _ {i} ( A) $
 +
are equal to the real parts of the eigen values of the operator  $  A ( 0) $(
 +
the matrix  $  \| a _ {j}  ^ {i} \| $).
  
1) A system with constant coefficients (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113039.png" />). In this case the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113040.png" /> are equal to the real parts of the eigen values of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113041.png" /> (the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113042.png" />).
+
2) A system with periodic coefficients (that is, $  A ( t + T ) \equiv A ( t) $,  
 
+
$  T > 0 $).  
2) A system with periodic coefficients (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113044.png" />). In this case
+
In this case
  
<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/l/l061/l061130/l06113045.png" /></td> </tr></table>
+
$$
 +
\lambda _ {i} ( A)  =
 +
\frac{1}{T}
 +
  \mathop{\rm ln}  | \mu _ {i} | ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113046.png" /> are the [[Multipliers|multipliers]] of the system (1), numbered in non-increasing order of their moduli (each is taken as many times as its multiplicity).
+
where $  \mu _ {i} $
 +
are the [[Multipliers|multipliers]] of the system (1), numbered in non-increasing order of their moduli (each is taken as many times as its multiplicity).
  
The role of the Lyapunov characteristic exponent in the theory of Lyapunov stability is based on the following assertion: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113047.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113048.png" />), then the solutions of (1) are asymptotically stable (respectively, unstable, cf. [[Asymptotically-stable solution|Asymptotically-stable solution]]). From <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113049.png" /> it does not follow that the null solution of the system
+
The role of the Lyapunov characteristic exponent in the theory of Lyapunov stability is based on the following assertion: If $  \lambda _ {1} ( A) < 0 $(
 +
> 0 $),  
 +
then the solutions of (1) are asymptotically stable (respectively, unstable, cf. [[Asymptotically-stable solution|Asymptotically-stable solution]]). From $  \lambda _ {1} ( A) < 0 $
 +
it does not follow that the null solution 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/l/l061/l061130/l06113050.png" /></td> </tr></table>
+
$$
 +
\dot{x}  = A ( t) x + O ( | x |  ^ {2} )
 +
$$
  
 
is Lyapunov stable; however, if it is also known that the system (1) is a [[Regular linear system|regular linear system]], then this conclusion is valid (Lyapunov's theorem).
 
is Lyapunov stable; however, if it is also known that the system (1) is a [[Regular linear system|regular linear system]], then this conclusion is valid (Lyapunov's theorem).
  
Suppose that the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113051.png" /> is obtained by a small perturbation of a system (1) satisfying the condition
+
Suppose that the system $  \dot{x} = B ( t) x $
 +
is obtained by a small perturbation of a system (1) 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/l/l061/l061130/l06113052.png" /></td> </tr></table>
+
$$
 +
\sup _ {t \in \mathbf R }  \| A ( t) \|  < + \infty ;
 +
$$
  
 
that is, the distance between them, defined by the formula
 
that is, the distance between them, defined by the formula
  
<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/l/l061/l061130/l06113053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
d ( A , B )  = \
 +
\sup _ {t \in \mathbf R }  \| A ( t) - ( t) \| ,
 +
$$
  
is small. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113054.png" /> this does not imply that the quantity
+
is small. For $  n > 1 $
 +
this does not imply that the quantity
  
<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/l/l061/l061130/l06113055.png" /></td> </tr></table>
+
$$
 +
| \lambda _ {1} ( A) - \lambda _ {1} ( B) |
 +
$$
  
is small (it is implied if the system (1) has constant or periodic coefficients, and also for certain other systems); in other words, the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113056.png" /> are not everywhere continuous on the space of systems (1) (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113057.png" />), endowed with the given metric (2).
+
is small (it is implied if the system (1) has constant or periodic coefficients, and also for certain other systems); in other words, the functionals $  \lambda _ {i} ( A) $
 +
are not everywhere continuous on the space of systems (1) ( $  \sup _ {t \in \mathbf R }  \| A ( t) \| < + \infty $),  
 +
endowed with the given metric (2).
  
Lyapunov characteristic exponents were introduced by A.M. Lyapunov, not only for solutions of the system (1), but also for arbitrary functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113058.png" /> (see [[#References|[1]]]).
+
Lyapunov characteristic exponents were introduced by A.M. Lyapunov, not only for solutions of the system (1), but also for arbitrary functions on $  \mathbf R  ^ {+} $(
 +
see [[#References|[1]]]).
  
 
====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">  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">[3]</TD> <TD valign="top">  N.A. Izobov,  "Linear systems of ordinary differential equations"  ''J. Soviet Math.'' , '''5'''  (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">  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">[3]</TD> <TD valign="top">  N.A. Izobov,  "Linear systems of ordinary differential equations"  ''J. Soviet Math.'' , '''5'''  (1976)  pp. 46–96  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''12'''  (1974)  pp. 71–146</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Presently, Lyapunov (characteristic) exponents are used at a much wider scale. For a survey see [[#References|[a6]]]. First, the matrix A may be a stochastic time-dependent function. Lyapunov exponents are also used in relation with a system of non-linear differential equations
 
Presently, Lyapunov (characteristic) exponents are used at a much wider scale. For a survey see [[#References|[a6]]]. First, the matrix A may be a stochastic time-dependent function. Lyapunov exponents are also used in relation with a system of non-linear 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/l/l061/l061130/l06113059.png" /></td> </tr></table>
+
$$
 +
= f ( x) ,\  x = ( x  ^ {1} ( t) \dots x  ^ {n} ( t) ),
 +
$$
  
having a [[Strange attractor|strange attractor]] (or repellor) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113060.png" /> as limit solution, see [[#References|[a7]]]. The system linearized in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113061.png" /> is of the form (1) with
+
having a [[Strange attractor|strange attractor]] (or repellor) $  s ( t) $
 +
as limit solution, see [[#References|[a7]]]. The system linearized in $  s ( t) $
 +
is of the form (1) with
  
<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/l/l061/l061130/l06113062.png" /></td> </tr></table>
+
$$
 +
A ( t)  = \left \{
 +
\frac{\partial  f _ {i} }{\partial  x _ {j} }
 +
( s( t) ) \right
 +
\} _ {( n \times n ) }  .
 +
$$
  
One of the exponents is zero. The occurrence of one or more positive exponents indicates that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113063.png" /> is a strange attractor. For a conservative system the sum of the Lyapunov exponents is zero, while for a dissipative system the sum is negative. The capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113064.png" /> of a strange attractor is a fractal number related to the [[Hausdorff dimension|Hausdorff dimension]]. J.L. Kaplan and J.A. Yorke made the conjecture
+
One of the exponents is zero. The occurrence of one or more positive exponents indicates that $  s ( t) $
 +
is a strange attractor. For a conservative system the sum of the Lyapunov exponents is zero, while for a dissipative system the sum is negative. The capacity $  D $
 +
of a strange attractor is a fractal number related to the [[Hausdorff dimension|Hausdorff dimension]]. J.L. Kaplan and J.A. Yorke made the conjecture
  
<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/l/l061/l061130/l06113065.png" /></td> </tr></table>
+
$$
 +
= j -  
 +
\frac{1}{\lambda _ {j+} 1 }
 +
\sum _ { i= } 1 ^ { j }  \lambda _ {i}  $$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113066.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113067.png" />).
+
with $  0 < \sum _ {i=} 1  ^ {j} \lambda _ {i} < - \lambda _ {j+} 1 $(
 +
$  \lambda _ {1} \geq  \dots \geq  \lambda _ {n} $).
  
 
The concept of Lyapunov exponents extends to non-linear stochastic systems, as well as to iteration mappings
 
The concept of Lyapunov exponents extends to non-linear stochastic systems, as well as to iteration mappings
  
<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/l/l061/l061130/l06113068.png" /></td> </tr></table>
+
$$
 +
x ( t + 1 )  = A ( t) x ( t) ,
 +
$$
  
 
see [[#References|[a6]]] and [[#References|[a7]]].
 
see [[#References|[a6]]] and [[#References|[a7]]].
  
In a yet more general deterministic setting, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113069.png" /> be a compact subset of a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113070.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113071.png" /> be a mapping such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113072.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113073.png" /> is supposed to satisfy the following uniform differentiability condition: For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113074.png" /> there is a linear compact operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113075.png" /> such that
+
In a yet more general deterministic setting, let $  X $
 +
be a compact subset of a Hilbert space $  H $,  
 +
and let $  f : X \rightarrow H $
 +
be a mapping such that $  f ( X) = X $.  
 +
The mapping $  f $
 +
is supposed to satisfy the following uniform differentiability condition: For each $  x \in X $
 +
there is a linear compact operator $  L ( x) : H \rightarrow H $
 +
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/l/l061/l061130/l06113076.png" /></td> </tr></table>
+
$$
 +
\sup _ {x,y } 
 +
\frac{\| f ( y) - f ( x) -
 +
L ( x) ( y - x ) \| }{\| y - x \| }
 +
  \rightarrow  0
 +
$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113077.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113078.png" /> denotes the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113079.png" />, and where the supremum is taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113080.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113081.png" />.
+
as $  \epsilon \rightarrow 0 $,  
 +
where $  \| \cdot \| $
 +
denotes the norm in $  H $,  
 +
and where the supremum is taken over all $  x , y $
 +
with  $  0 < \| x - y \| \leq  \epsilon $.
  
For a compact linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113082.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113083.png" /> be the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113084.png" />. For each positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113085.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113086.png" />, while for a non-integer positive real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113088.png" />, define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113089.png" />.
+
For a compact linear operator $  L $,  
 +
let $  \alpha _ {1} ( L) \geq  \alpha _ {2} ( L) \geq  \dots $
 +
be the eigen values of $  ( L  ^ {*} L )  ^ {1/2} $.  
 +
For each positive integer $  d $,  
 +
let $  \omega _ {d} ( L) = \alpha _ {1} ( L) \dots \alpha _ {d} ( L) $,  
 +
while for a non-integer positive real number $  d = n + s $,
 +
0 < s < 1 $,  
 +
define $  \omega _ {d} ( L) = \omega _ {n} ( L) ( \alpha _ {n+} 1 ( L) )  ^ {s} $.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113091.png" /> are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113092.png" />.
+
Suppose that $  f $
 +
and $  L $
 +
are such that $  \sup _ {x \in X }  \| L ( x) \| < \infty $.
  
For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113093.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113094.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113095.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113096.png" />-th iterate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l06113097.png" />. Define the (local) Lyapunov numbers and Lyapunov exponents by
+
For each $  x \in X $,  
 +
let $  L _ {p} ( x) = L ( f ^ { p- 1 } ( x) ) \circ \dots L ( f ( x) ) \circ L ( x) $,  
 +
where $  f ^ { r } $
 +
denotes the $  r $-
 +
th iterate of $  f $.  
 +
Define the (local) Lyapunov numbers and Lyapunov exponents by
  
<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/l/l061/l061130/l06113098.png" /></td> </tr></table>
+
$$
 +
\Lambda _ {j} ( x)  = \lim\limits  \sup _ {p \rightarrow \infty }  \{ \alpha _ {j} ( L _ {p} ( x) ) \}  ^ {1/p} ,
 +
$$
  
<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/l/l061/l061130/l06113099.png" /></td> </tr></table>
+
$$
 +
\mu _ {j} ( x)  =   \mathop{\rm ln}  \Lambda _ {j} ( x) .
 +
$$
  
The uniform Lyapunov exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l061130100.png" /> and uniform Lyapunov numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l061130101.png" /> in this setting are defined as follows:
+
The uniform Lyapunov exponents $  \mu _ {i} $
 +
and uniform Lyapunov numbers $  \Lambda _ {i} $
 +
in this setting are defined as follows:
  
<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/l/l061/l061130/l061130102.png" /></td> </tr></table>
+
$$
 +
\overline \omega \; _ {j} ( p)  = \sup _ {x \in X }  \omega _ {j} ( L _ {p} ( x) ) ,
 +
$$
  
<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/l/l061/l061130/l061130103.png" /></td> </tr></table>
+
$$
 +
\Pi _ {j}  = \lim\limits _ {p \rightarrow \infty }  ( \overline \omega \; _ {j} ( p) )  ^ {1/p} ,
 +
$$
  
<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/l/l061/l061130/l061130104.png" /></td> </tr></table>
+
$$
 +
\Lambda _ {1} \dots \Lambda _ {j}  = \Pi _ {j} ,\  j = 1 , 2 \dots
 +
$$
  
<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/l/l061/l061130/l061130105.png" /></td> </tr></table>
+
$$
 +
\mu _ {j}  =   \mathop{\rm ln}  \Lambda _ {j} ,\  j = 1 , 2 ,\dots .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l061130106.png" /> be the smallest integer such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l061130107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l061130108.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l061130109.png" /> is then called the Lyapunov dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l061130110.png" />. One has (see [[#References|[a2]]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l061130111.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l061130112.png" /> is the [[Hausdorff dimension|Hausdorff dimension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061130/l061130113.png" />.
+
Let $  n $
 +
be the smallest integer such that $  \mu _ {1} + \dots + \mu _ {n} \geq  0 $
 +
and $  \mu _ {1} + \dots + \mu _ {n} + \mu _ {n+} 1 < 0 $.  
 +
The number $  d _ {L} ( X) = n + | \mu _ {n+} 1 |  ^ {-} 1 ( \mu _ {1} + \dots + \mu _ {n} ) $
 +
is then called the Lyapunov dimension of $  X $.  
 +
One has (see [[#References|[a2]]]) $  d _ {H} ( X) \leq  d _ {L} ( X) $,  
 +
where $  d _ {H} ( X) $
 +
is the [[Hausdorff dimension|Hausdorff dimension]] of $  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Kliemann,  "Analysis of nonlinear stochastic systems"  W. Schiehlen (ed.)  W. Wedig (ed.) , ''Analysis and estimation of stochastic mechanical systems'' , Springer (Wien)  (1988)  pp. 43–102</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Constantin,  C. Foias,  R. Temam,  "Attractors representing turbulent flows" , Amer. Math. Soc.  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.S. Young,  "Capacity of attractors"  ''Ergod. Th. Dynam. Systems'' , '''1'''  (1981)  pp. 381–383</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.S. Young,  "Dimension, entropy, and Lyapunov exponents"  ''Ergod. Th. Dynam. Systems'' , '''2'''  (1982)  pp. 109–124</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Fredrickson,  J.L. Kaplan,  E.D. Yorke,  J.A. Yorke,  "The Lyapunov dimension of strange attractors"  ''J. Diff. Eq.'' , '''49'''  (1983)  pp. 185–207</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Arnold (ed.)  V. Wihstutz (ed.) , ''Lyapunov exponents'' , ''Lect. notes in math.'' , '''1186''' , Springer  (1986)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H.G. Schuster,  "Deterministic chaos" , Physik-Verlag  (1988)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J. Guckenheimer,  P. Holmes,  "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Kliemann,  "Analysis of nonlinear stochastic systems"  W. Schiehlen (ed.)  W. Wedig (ed.) , ''Analysis and estimation of stochastic mechanical systems'' , Springer (Wien)  (1988)  pp. 43–102</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Constantin,  C. Foias,  R. Temam,  "Attractors representing turbulent flows" , Amer. Math. Soc.  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.S. Young,  "Capacity of attractors"  ''Ergod. Th. Dynam. Systems'' , '''1'''  (1981)  pp. 381–383</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.S. Young,  "Dimension, entropy, and Lyapunov exponents"  ''Ergod. Th. Dynam. Systems'' , '''2'''  (1982)  pp. 109–124</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Fredrickson,  J.L. Kaplan,  E.D. Yorke,  J.A. Yorke,  "The Lyapunov dimension of strange attractors"  ''J. Diff. Eq.'' , '''49'''  (1983)  pp. 185–207</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Arnold (ed.)  V. Wihstutz (ed.) , ''Lyapunov exponents'' , ''Lect. notes in math.'' , '''1186''' , Springer  (1986)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H.G. Schuster,  "Deterministic chaos" , Physik-Verlag  (1988)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J. Guckenheimer,  P. Holmes,  "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer  (1983)</TD></TR></table>

Latest revision as of 04:11, 6 June 2020


of a solution of a linear system

The limes superior

$$ \lambda _ {x ( t) } = \ {\lim\limits _ {t \rightarrow + \infty } } bar \ \frac{1}{t} \ \mathop{\rm ln} | x ( t) | , $$

where $ x ( t) \neq 0 $ is a solution of the linear system of ordinary differential equations

$$ \tag{1 } \dot{x} = A ( t) x ; $$

here $ x \in \mathbf R ^ {n} $ and $ A ( \cdot ) $ is a mapping $ \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $, summable on every interval. In coordinates,

$$ x ( t) = ( x ^ {1} ( t) \dots x ^ {n} ( t) ) , $$

$$ \dot{x} ^ {i} = \sum _ { j= } 1 ^ { n } a _ {j} ^ {i} ( t) x ^ {j} ,\ i = 1 \dots n , $$

where $ a _ {j} ^ {i} ( t) $ are functions summable on every interval and

$$ | x | = \sqrt {\sum _ { i= } 1 ^ { n } | x ^ {i} | ^ {2} } $$

(or any other equivalent norm; $ \lambda _ {x ( t) } $ does not depend on the choice of the norm in $ \mathbf R ^ {n} $ or in $ \mathbf C ^ {n} $).

Lyapunov's theorem. Suppose that

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

equivalently:

$$ {\lim\limits _ {t \rightarrow + \infty } } bar \ \frac{1}{t} \int\limits _ { 0 } ^ { t } | a _ {j} ^ {i} ( \tau ) | d \tau < + \infty ,\ \ i , j = 1 \dots n . $$

Then for any solution $ x ( t) \neq 0 $ of the system (1) the Lyapunov characteristic exponent $ \lambda _ {x ( t) } $ is a real number (that is, $ \neq \pm \infty $). The following assertions hold for the Lyapunov characteristic exponents of non-zero solutions of (1):

1) $ \lambda _ {\alpha x ( t) } = \lambda _ {x ( t) } $, $ \alpha \neq 0 $;

2) $ \lambda _ {x _ {1} ( t) + x _ {2} ( t) } \leq \max ( \lambda _ {x _ {1} ( t) } , \lambda _ {x _ {2} ( t) } ) $;

3) there exists a set of linearly independent solutions of (1), denoted by $ \{ x _ {i} ( t) \} _ {i=} 1 ^ {n} $, such that for any other $ n $ linearly independent solutions $ \widehat{x} _ {i} ( t) $, $ i = 1 \dots n $, of (1), numbered in decreasing order of the Lyapunov characteristic exponents, that is, $ \lambda _ {\widehat{x} _ {i} ( t) } \geq \lambda _ {\widehat{x} _ {j} ( t) } $ for $ i \leq j $, the following inequalities hold:

$$ \lambda _ {\widehat{x} _ {i} ( t) } \geq \ \lambda _ {x _ {i} ( t) } ,\ \ i = 1 \dots n . $$

A fundamental system of solutions $ \{ x _ {i} ( t) \} _ {i=} 1 ^ {n} $ with this property is called normal. Such a normal system has the properties:

a) the family of numbers $ \lambda _ {i} ( A) = \lambda _ {x _ {i} ( t) } $, $ i = 1 \dots n $, does not depend on the choice of the normal fundamental system;

b) for any solution $ x ( t) \neq 0 $ of (1) its Lyapunov characteristic exponent $ \lambda _ {x ( t) } $ is equal to some $ \lambda _ {i} ( A) $;

c) $ \lambda _ {i} ( A) \geq \lambda _ {j} ( A) $, $ i \leq j $.

The numbers $ \lambda _ {1} ( A) \geq \dots \geq \lambda _ {n} ( A) $ are called the Lyapunov characteristic exponents of the system (1); the number $ \lambda _ {1} ( A) $ is often called the leading Lyapunov characteristic exponent of (1).

The set of all Lyapunov characteristic exponents of non-zero solutions of (1) is called the spectrum.

Special case.

1) A system with constant coefficients (that is, $ A ( t) \equiv A ( 0) $). In this case the $ \lambda _ {i} ( A) $ are equal to the real parts of the eigen values of the operator $ A ( 0) $( the matrix $ \| a _ {j} ^ {i} \| $).

2) A system with periodic coefficients (that is, $ A ( t + T ) \equiv A ( t) $, $ T > 0 $). In this case

$$ \lambda _ {i} ( A) = \frac{1}{T} \mathop{\rm ln} | \mu _ {i} | , $$

where $ \mu _ {i} $ are the multipliers of the system (1), numbered in non-increasing order of their moduli (each is taken as many times as its multiplicity).

The role of the Lyapunov characteristic exponent in the theory of Lyapunov stability is based on the following assertion: If $ \lambda _ {1} ( A) < 0 $( $ > 0 $), then the solutions of (1) are asymptotically stable (respectively, unstable, cf. Asymptotically-stable solution). From $ \lambda _ {1} ( A) < 0 $ it does not follow that the null solution of the system

$$ \dot{x} = A ( t) x + O ( | x | ^ {2} ) $$

is Lyapunov stable; however, if it is also known that the system (1) is a regular linear system, then this conclusion is valid (Lyapunov's theorem).

Suppose that the system $ \dot{x} = B ( t) x $ is obtained by a small perturbation of a system (1) satisfying the condition

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

that is, the distance between them, defined by the formula

$$ \tag{2 } d ( A , B ) = \ \sup _ {t \in \mathbf R } \| A ( t) - B ( t) \| , $$

is small. For $ n > 1 $ this does not imply that the quantity

$$ | \lambda _ {1} ( A) - \lambda _ {1} ( B) | $$

is small (it is implied if the system (1) has constant or periodic coefficients, and also for certain other systems); in other words, the functionals $ \lambda _ {i} ( A) $ are not everywhere continuous on the space of systems (1) ( $ \sup _ {t \in \mathbf R } \| A ( t) \| < + \infty $), endowed with the given metric (2).

Lyapunov characteristic exponents were introduced by A.M. Lyapunov, not only for solutions of the system (1), but also for arbitrary functions on $ \mathbf R ^ {+} $( see [1]).

References

[1] A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[2] 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)
[3] N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 (1976) pp. 46–96 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 71–146

Comments

Presently, Lyapunov (characteristic) exponents are used at a much wider scale. For a survey see [a6]. First, the matrix A may be a stochastic time-dependent function. Lyapunov exponents are also used in relation with a system of non-linear differential equations

$$ x = f ( x) ,\ x = ( x ^ {1} ( t) \dots x ^ {n} ( t) ), $$

having a strange attractor (or repellor) $ s ( t) $ as limit solution, see [a7]. The system linearized in $ s ( t) $ is of the form (1) with

$$ A ( t) = \left \{ \frac{\partial f _ {i} }{\partial x _ {j} } ( s( t) ) \right \} _ {( n \times n ) } . $$

One of the exponents is zero. The occurrence of one or more positive exponents indicates that $ s ( t) $ is a strange attractor. For a conservative system the sum of the Lyapunov exponents is zero, while for a dissipative system the sum is negative. The capacity $ D $ of a strange attractor is a fractal number related to the Hausdorff dimension. J.L. Kaplan and J.A. Yorke made the conjecture

$$ D = j - \frac{1}{\lambda _ {j+} 1 } \sum _ { i= } 1 ^ { j } \lambda _ {i} $$

with $ 0 < \sum _ {i=} 1 ^ {j} \lambda _ {i} < - \lambda _ {j+} 1 $( $ \lambda _ {1} \geq \dots \geq \lambda _ {n} $).

The concept of Lyapunov exponents extends to non-linear stochastic systems, as well as to iteration mappings

$$ x ( t + 1 ) = A ( t) x ( t) , $$

see [a6] and [a7].

In a yet more general deterministic setting, let $ X $ be a compact subset of a Hilbert space $ H $, and let $ f : X \rightarrow H $ be a mapping such that $ f ( X) = X $. The mapping $ f $ is supposed to satisfy the following uniform differentiability condition: For each $ x \in X $ there is a linear compact operator $ L ( x) : H \rightarrow H $ such that

$$ \sup _ {x,y } \frac{\| f ( y) - f ( x) - L ( x) ( y - x ) \| }{\| y - x \| } \rightarrow 0 $$

as $ \epsilon \rightarrow 0 $, where $ \| \cdot \| $ denotes the norm in $ H $, and where the supremum is taken over all $ x , y $ with $ 0 < \| x - y \| \leq \epsilon $.

For a compact linear operator $ L $, let $ \alpha _ {1} ( L) \geq \alpha _ {2} ( L) \geq \dots $ be the eigen values of $ ( L ^ {*} L ) ^ {1/2} $. For each positive integer $ d $, let $ \omega _ {d} ( L) = \alpha _ {1} ( L) \dots \alpha _ {d} ( L) $, while for a non-integer positive real number $ d = n + s $, $ 0 < s < 1 $, define $ \omega _ {d} ( L) = \omega _ {n} ( L) ( \alpha _ {n+} 1 ( L) ) ^ {s} $.

Suppose that $ f $ and $ L $ are such that $ \sup _ {x \in X } \| L ( x) \| < \infty $.

For each $ x \in X $, let $ L _ {p} ( x) = L ( f ^ { p- 1 } ( x) ) \circ \dots L ( f ( x) ) \circ L ( x) $, where $ f ^ { r } $ denotes the $ r $- th iterate of $ f $. Define the (local) Lyapunov numbers and Lyapunov exponents by

$$ \Lambda _ {j} ( x) = \lim\limits \sup _ {p \rightarrow \infty } \{ \alpha _ {j} ( L _ {p} ( x) ) \} ^ {1/p} , $$

$$ \mu _ {j} ( x) = \mathop{\rm ln} \Lambda _ {j} ( x) . $$

The uniform Lyapunov exponents $ \mu _ {i} $ and uniform Lyapunov numbers $ \Lambda _ {i} $ in this setting are defined as follows:

$$ \overline \omega \; _ {j} ( p) = \sup _ {x \in X } \omega _ {j} ( L _ {p} ( x) ) , $$

$$ \Pi _ {j} = \lim\limits _ {p \rightarrow \infty } ( \overline \omega \; _ {j} ( p) ) ^ {1/p} , $$

$$ \Lambda _ {1} \dots \Lambda _ {j} = \Pi _ {j} ,\ j = 1 , 2 \dots $$

$$ \mu _ {j} = \mathop{\rm ln} \Lambda _ {j} ,\ j = 1 , 2 ,\dots . $$

Let $ n $ be the smallest integer such that $ \mu _ {1} + \dots + \mu _ {n} \geq 0 $ and $ \mu _ {1} + \dots + \mu _ {n} + \mu _ {n+} 1 < 0 $. The number $ d _ {L} ( X) = n + | \mu _ {n+} 1 | ^ {-} 1 ( \mu _ {1} + \dots + \mu _ {n} ) $ is then called the Lyapunov dimension of $ X $. One has (see [a2]) $ d _ {H} ( X) \leq d _ {L} ( X) $, where $ d _ {H} ( X) $ is the Hausdorff dimension of $ X $.

References

[a1] W. Kliemann, "Analysis of nonlinear stochastic systems" W. Schiehlen (ed.) W. Wedig (ed.) , Analysis and estimation of stochastic mechanical systems , Springer (Wien) (1988) pp. 43–102
[a2] P. Constantin, C. Foias, R. Temam, "Attractors representing turbulent flows" , Amer. Math. Soc. (1985)
[a3] L.S. Young, "Capacity of attractors" Ergod. Th. Dynam. Systems , 1 (1981) pp. 381–383
[a4] L.S. Young, "Dimension, entropy, and Lyapunov exponents" Ergod. Th. Dynam. Systems , 2 (1982) pp. 109–124
[a5] P. Fredrickson, J.L. Kaplan, E.D. Yorke, J.A. Yorke, "The Lyapunov dimension of strange attractors" J. Diff. Eq. , 49 (1983) pp. 185–207
[a6] L. Arnold (ed.) V. Wihstutz (ed.) , Lyapunov exponents , Lect. notes in math. , 1186 , Springer (1986)
[a7] H.G. Schuster, "Deterministic chaos" , Physik-Verlag (1988)
[a8] E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[a9] J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)
How to Cite This Entry:
Lyapunov characteristic exponent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lyapunov_characteristic_exponent&oldid=12994
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article