View source for Conditional stability

<!--
c0245201.png
$#A+1 = 225 n = 0
$#C+1 = 225 : ~/encyclopedia/old_files/data/C024/C.0204520 Conditional stability
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}}

''of a point relative to a family of mappings

$$ \tag{1 }
\{ f _ {t} \} _ {f \in G  ^ {+}  } : \ 
E  \rightarrow  E
$$

''

Equicontinuity at this point of the family  $  \{ f _ {t} \mid  _ {V} \} _ {t \in G  ^ {+}  } $
of restrictions of the mappings  $  f _ {t} $
to a certain manifold  $  V $
imbedded in  $  E $(
with the induced metric on  $  V $); 
here  $  G  ^ {+} $
is the set of real or integer non-negative numbers:  $  G = \mathbf R $
or  $  G = \mathbf Z $.

The conditional stability of a point relative to a mapping is defined as the conditional stability relative to the family of non-negative powers of this mapping. The conditional stability of a point relative to a dynamical system  $  f ^ { t } $
is the conditional stability of this point relative to the family of mappings  $  \{ f ^ { t } \} _ {t \in G  ^ {+}  } $. 
The conditional stability of a solution of an equation

$$ 
x ( t + 1 )  = \ 
g _ {t} x ( t)
$$

given on  $  t _ {0} + \mathbf Z  ^ {+} $
is the conditional stability of the point  $  x _ {0} ( t _ {0} ) $
relative to the family of mappings

$$ 
\left \{
f _ {t}  \stackrel{\rm def}{=}\ 
g _ {t _ {0}  + t } \dots g _ {t _ {0}  + 1 } g _ {t _ {0}  }
\right \} _ {t \in \mathbf Z  ^ {+}  } .
$$

The conditional stability of the solution  $  x _ {0} ( \cdot ) $
of a differential equation

$$ \tag{2 }
\dot{x}  =  f ( x , t )
$$

given on  $  t _ {0} + \mathbf R  ^ {+} $
is the conditional stability of the point  $  x _ {0} ( t _ {0} ) $
relative to the family of mappings  $  \{ X ( t _ {0} + t , t _ {0} ) \} _ {t \in \mathbf R  ^ {+}  } $, 
where  $  X ( \theta , \tau ) $
is the [[Cauchy operator|Cauchy operator]] of this equation. The conditional stability of the solution  $  y ( \cdot ) $
of a differential equation of order  $  m $,

$$ 
y  ^ {(} m)  =  g ( y , \dot{y} \dots y  ^ {(} m- 1) , t )
$$

given on  $  t _ {0} + \mathbf R  ^ {+} $, 
is the conditional stability of the solution  $  x ( \cdot ) = ( y ( \cdot ) , \dot{y} ( \cdot ) \dots y  ^ {(} m- 1) ( \cdot ) ) $, 
given on  $  t _ {0} + \mathbf R  ^ {+} $, 
of the corresponding first-order differential equation of the form (2), where

$$ 
x  =  ( x _ {1} \dots x _ {m} ) ,
$$

$$ 
f ( x , t )  =  ( x _ {2} \dots x _ {m} , g ( x _ {1} \dots x _ {m} , t ) ).
$$

The definitions 1)–5) below are some concrete examples of these and related notions.

1) Given a differential equation (2), where  $  E $
is a normed  $  n $-
dimensional vector space and  $  x \in E $. 
The solution  $  x _ {0} ( \cdot ) :  t _ {0} + \mathbf R  ^ {+} \rightarrow E $
of this equation is called conditionally stable with index  $  k \in \{ 0 \dots n \} $
if there is a  $  k $-
dimensional disc  $  D  ^ {k} $
imbedded in  $  E $(
considered as a manifold of class  $  C  ^ {m} $), 
containing the point  $  x _ {0} ( t _ {0} ) $
and having the following property: For each  $  \epsilon > 0 $
there is a  $  \delta > 0 $
such that for every  $  x \in D  ^ {k} $
satisfying the inequality  $  | x - x _ {0} ( t _ {0} ) | < \delta $, 
the solution  $  x ( \cdot ) $
of the same equation satisfying the initial condition  $  x ( t _ {0} ) = x $
is uniquely defined on  $  t _ {0} + \mathbf R  ^ {+} $, 
and for each  $  t \in t _ {0} + \mathbf R  ^ {+} $
satisfies the inequality  $  | x ( t) - x _ {0} ( t) | < \epsilon $. 
If the disc  $  D  ^ {k} $
with the given property may be chosen so that

$$ 
\lim\limits _
{t \rightarrow + \infty } \ 
| x ( t) - x _ {0} ( t) |  =  0
$$

(respectively,

$$ 
{\lim\limits _ {t \rightarrow + \infty } } bar \ 

\frac{1}{t}
   \mathop{\rm ln}  | x ( t) - x _ {0} ( t) |  <  0 ;
$$

here, and elsewhere is understood that  $   \mathop{\rm ln}  0 = - \infty $) 
for every solution of the same equations starting in this disc (i.e. such that  $  x ( t _ {0} ) \in D  ^ {k} $), 
then the solution  $  x _ {0} ( t) $
is called asymptotically (respectively, exponentially) conditionally stable (with index  $  k $).

The solution of the equation (2) ( $  x \in \mathbf R  ^ {n} $
or  $  x \in \mathbf C  ^ {n} $) 
is called conditionally (asymptotically, exponentially conditionally) stable with index  $  k $, 
if it becomes such as the result of equipping  $  \mathbf R  ^ {n} $(
or  $  \mathbf C  ^ {n} $) 
with a suitable norm. This property of the solution does not depend on the choice of norm.

2) Given an  $  n $-
dimensional Riemannian manifold  $  V  ^ {n} $(
the distance function on which is written as  $  d ( \cdot , \cdot ) $), 
one calls a point  $  x _ {0} \in V  ^ {n} $
conditionally stable (with index  $  k \in \{ 0 \dots n \} $) 
relative to a mapping  $  f :  V  ^ {n} \rightarrow V  ^ {n} $
if there is a (usually smooth)  $  k $-
dimensional disc  $  D  ^ {k} $
imbedded in  $  V  ^ {n} $, 
containing the point  $  x _ {0} $
and having the following property: For each  $  \epsilon > 0 $
there exists a  $  \delta > 0 $
such that for every  $  x \in D  ^ {k} $
satisfying the inequality  $  d ( x , x _ {0} ) < \delta $, 
the inequality  $  d ( f ^ { t } x , f ^ { t } x _ {0} ) < \epsilon $
holds for all  $  t \in \mathbf N $. 
If the disc  $  D  ^ {k} $
with the above property can be chosen so that

$$ 
d ( f ^ { t } x , f ^ { t } x _ {0} )  \rightarrow  0 \ \ 
\textrm{ as }  t \rightarrow + \infty
$$

(respectively,

$$ 
\left . {\lim\limits _ {t \rightarrow + \infty } } bar \ 

\frac{1}{t}
   \mathop{\rm ln} \ 
d ( f ^ { t } x , f ^ { t } x _ {0} )  <  0 \right )
$$

for each  $  x \in D  ^ {k} $, 
then the point  $  x _ {0} $
is called asymptotically (respectively, exponentially) conditionally stable (with index  $  k $) 
relative to the mapping  $  f $.

Let  $  V  ^ {n} $
be a compact differentiable manifold. A point  $  x _ {0} \in V  ^ {n} $
is called conditionally stable (asymptotically, exponentially conditionally stable) with index  $  k $
relative to a mapping  $  f :  V  ^ {n} \rightarrow V  ^ {n} $
if it becomes such as a result of equipping  $  V  ^ {n} $
with a suitable Riemannian metric. This property of  $  x _ {0} $
does not depend on the choice of the Riemannian metric on  $  V  ^ {n} $.

3) Consider the differential equation (2) on an  $  n $-
dimensional Riemannian (or Finsler, cf. [[Finsler geometry|Finsler geometry]]) manifold  $  V  ^ {n} $, 
the distance function on which is denoted by  $  d ( \cdot , \cdot ) $. 
The solution  $  x _ {0} ( \cdot ) :  t _ {0} + \mathbf R  ^ {+} \rightarrow V  ^ {n} $
of this equation is called conditionally stable (with index  $  k $) 
if there is a  $  k $-
dimensional disc  $  D  ^ {k} $
imbedded in  $  V  ^ {n} $(
considered as a manifold of class  $  C  ^ {m} $, 
where usually  $  m \geq  1 $), 
containing the point  $  x _ {0} ( t _ {0} ) $
and having the following property: For each  $  \epsilon > 0 $
there exists a  $  \delta > 0 $
such that for every  $  x \in D  ^ {k} $
satisfying the inequality  $  d ( x , x _ {0} ( t _ {0} ) ) < \delta $, 
the solution  $  x ( \cdot ) $
of the same equation satisfying the initial condition  $  x ( t _ {0} ) = x $
is unique, defined on  $  t _ {0} + \mathbf R  ^ {+} $, 
and for each  $  t \in t _ {0} + \mathbf R  ^ {+} $
satisfies the inequality  $  d ( x ( t) , x _ {0} ( t) ) < \epsilon $. 
If the disc  $  D  ^ {k} $
with the above property may be taken so that

$$ 
d ( x ( t) , x _ {0} ( t) )  \rightarrow  0 \ \ 
\textrm{ as }  t \rightarrow + \infty
$$

(respectively,

$$ 
\left . {\lim\limits _ {t \rightarrow + \infty } } bar \ 

\frac{1}{t}
   \mathop{\rm ln}  d ( x ( t) , x _ {0} ( t) )  <  0 \right )
$$

for every solution of the same equation starting in this disc (i.e. such that  $  x ( t _ {0} ) \in D  ^ {k} $), 
then the solution  $  x _ {0} ( \cdot ) $
is called asymptotically (respectively, exponentially) conditionally stable (with index  $  k $).

4) Let  $  V  ^ {n} $
be an  $  n $-
dimensional manifold of class  $  C  ^ {m} $
and let  $  U $
be an open subset of it. Suppose that a point  $  x _ {0} \in U $
is fixed under a family of mappings  $  f _ {t} :  U \rightarrow V  ^ {n} $
of class  $  C  ^ {m} $(
$  t \in G  ^ {+} $, 
where  $  G $
is  $  \mathbf R $
or  $  \mathbf Z $). 
The fixed point  $  x _ {0} $
is called conditionally stable (with index  $  k $) 
relative to the family of mappings  $  \{ f _ {t} \} _ {t \in G  ^ {+}  } $
if there is  $  k $-
dimensional disc  $  D  ^ {k} $
smoothly imbedded (by an imbedding of class  $  C  ^ {m} $) 
in  $  V  ^ {n} $
such that for every neighbourhood  $  V \subset  V  ^ {n} $
of  $  x _ {0} $
there is a neighbourhood  $  W $
of the same point such that  $  f _ {t} ( D  ^ {k} \cap W ) \subset  V $
for every  $  t \in G  ^ {+} $. 
If the disc  $  D  ^ {k} $
with this property may be taken so that  $  \lim\limits _ {t \rightarrow + \infty }   f _ {t} x = x _ {0} $
for every  $  x \in D  ^ {k} $, 
then the fixed point  $  x _ {0} $
is called asymptotically conditionally stable (with index  $  k $) 
relative to the family of mappings  $  \{ f _ {t} \} _ {t \in G  ^ {+}  } $.

5) The conditional (conditional asymptotic, conditional exponential) stability (with index  $  k $) 
of the solution  $  y _ {0} ( \cdot ) $
of an equation of arbitrary order  $  y  ^ {(} m) = g ( y , \dot{y} \dots y  ^ {(} m- 1) , t ) $, 
is defined as the conditional (asymptotic, conditional exponential) stability (with index  $  k $) 
of the solution  $  x _ {0} ( \cdot ) = ( y _ {0} ( \cdot ) , \dot{y} _ {0} ( \cdot ) \dots y _ {0}  ^ {(} m- 1) ( \cdot )) $
of the corresponding first-order equation (2), where

$$ 
x  =  ( x _ {1} \dots x _ {m} ) ,
$$

$$ 
f ( x , t )  =  ( x _ {2} \dots x _ {m} , g ( x _ {1} \dots x _ {m} , t ) ) .
$$

Sometimes (cf. e.g. [[#References|[3]]]) in defining conditional stability one requires the index  $  k $
to be non-zero: conditional stability with index zero always holds. Conditional stability (conditional asymptotic, conditional exponential stability) with index  $  n $(
the dimension of the phase space) is the same as [[Lyapunov stability|Lyapunov stability]] (respectively, asymptotic, exponential stability).

The equilibrium positions under conditional stability have been investigated. Suppose that in a neighbourhood of a point  $  x _ {0} \in \mathbf R  ^ {n} $
an autonomous differential equation

$$ \tag{3 }
\dot{x}  =  f ( x)
$$

is given, the right-hand side of which is continuously differentiable and vanishes at the point  $  x _ {0} $. 
If in the open left half-plane in the complex plane there are  $  k $
eigen values of the derivative  $  d f _ {x _ {0}  } $, 
then this fixed point of equation (3) is conditionally exponentially stable with index  $  k $(
Lyapunov's theorem on conditional stability). For example, the upper equilibrium position  $  y = \pi $, 
$  \dot{y} = 0 $
of the equation of oscillation of a pendulum  $  \dot{y} dot = \omega  ^ {2}  \sin  y = 0 $
is exponentially conditionally stable with index 1, because one of the roots of the characteristic equation  $  \lambda  ^ {2} - \omega  ^ {2} = 0 $
of the variational equation (cf. [[Variational equations|Variational equations]])  $  \dot{y} dot - \omega  ^ {2} y = 0 $
is negative.

A fixed point  $  x _ {0} $
of a differentiable mapping  $  f :  \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $
is exponentially conditionally stable with index  $  k $
relative to  $  f $
if  $  k $
eigen values of the derivative  $  d f _ {x _ {0}  } $
lie in the open unit disc. A periodic point  $  x _ {0} $
of a differential mapping  $  f :  \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $, 
having period  $  m $, 
is conditionally (asymptotically conditionally, exponentially conditionally) stable with index  $  k $
relative to  $  f $
if and only if it has this property relative to  $  f ^ { m } $.

A periodic solution of an autonomous differential equation (3) with smooth right-hand side  $  f ( x) $
having period  $  T $
is (asymptotically, exponentially) conditionally stable with index  $  k $
if and only if its value at the point  $  t = 0 $
is (respectively, asymptotically, exponentially) conditionally stable with index  $  k $
relative to the mapping  $  X ( T , 0 ) $, 
where  $  X ( \theta , \tau ) $
is the Cauchy operator of (3).

The example of O. Perron (cf. [[Lyapunov stability|Lyapunov stability]]) shows that the negativity of  $  k $
Lyapunov exponents of the variational equation along the solution of (3) does not imply the conditional stability with index  $  k $
of this solution. However, one has the following theorem, which shows that the situation described be Perron's example is not generic.

1) Let  $  S $
be the set of all diffeomorphisms  $  f $
of a Euclidean space  $  E  ^ {n} $
having uniformly continuous derivatives satisfying the inequality

$$ 
\sup _
{x \in E  ^ {n} } \ 
\max
\{ \| d f _ {x} \| , \| ( d f _ {x} )  ^ {-} 1 \| \}
 <  + \infty .
$$

For every diffeomorphism  $  j \in S $
denote by  $  S _ {j} $
the set of diffeomorphisms  $  f \in S $
satisfying the inequality

$$ 
\sup _
{x \in E  ^ {n} } \ 
| f x - j x |
 <  + \infty ;
$$

on the set  $  S _ {j} $
the distance function

$$ 
d ( f , g )  = \ 
\sup _
{x \in E  ^ {n} } \ 
( | f x - g x | +
\| d f _ {x} - d g _ {x} \| )
$$

is given.

Fir each  $  j \in S $
one has in  $  S _ {j} \times E  ^ {n} $
an everywhere-dense set  $  D _ {j} $
of type  $  G _  \delta  $
with the following property: For every  $  ( f , x ) \in D _ {j} $
the point  $  x $
is exponentially conditionally stable relative to the diffeomorphism  $  f $
with index

$$ 
 \mathop{\rm dim} \ 
\left \{ {
\mathfrak r \in T _ {x} E  ^ {n} } : { {\lim\limits _ {m \rightarrow + \infty } } bar \ 

\frac{1}{m}
   \mathop{\rm ln}  | d f ^ { m } \mathfrak r | < 0
 } \right \}
,
$$

i.e. with index equal to the number of negative Lyapunov characteristic exponents of the variational equation (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]).

2) For a dynamical system given on a closed differentiable manifold, analogous theorems can be formulated in a way that is more simple and invariant from the point of view of differential topology. Let  $  V  ^ {n} $
be a closed differentiable manifold. The set  $  S $
of all diffeomorphisms  $  f $
of class  $  C  ^ {1} $
mapping  $  V  ^ {n} $
onto  $  V  ^ {n} $
is equipped with the  $  C  ^ {1} $-
topology. In the space  $  S \times V  ^ {n} $
there is an everywhere-dense set  $  D $
of type  $  G _  \delta  $
with the following property: For each  $  ( f , x ) \in D $
the point  $  x $
is exponentially conditionally stable relative to the diffeomorphism  $  f $
with index

$$ \tag{4 }
k ( x)  =   \mathop{\rm dim} \ 
\left \{ {
\mathfrak r \in T _ {x} V  ^ {n} } : { {\lim\limits _ {m \rightarrow + \infty } } bar \ 

\frac{1}{m}
   \mathop{\rm ln}  | d f ^ { m } \mathfrak r | < 0
 } \right \}
.
$$

3) For every diffeomorphism  $  f :  V  ^ {n} \rightarrow V  ^ {n} $
of a closed differentiable manifold  $  V  ^ {n} $
and for every probability distribution on  $  V  ^ {n} $
that is invariant relative to  $  f $(
and the  $  \sigma $-
algebra of which contains all Borel sets), the set of points  $  x \in V  ^ {n} $
that are exponentially conditionally stable with index (4) relative to  $  f $
has probability 1.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,   "Collected works" , '''2''' , Moscow-Leningrad  (1956)  (In 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">  B.P. Demidovich,   "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.A. Izobov,   "Linear systems of ordinary differential equations"  ''J. Soviet Math.'' , '''5''' :  1  pp. 46–96  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''12'''  (1974)  pp. 71–146</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Ya.B. Pesin,   "Characteristic Lyapunov exponents and smooth ergodic theory"  ''Russian Math. Surveys'' , '''32''' :  4  (1977)  pp. 55–114  ''Uspekhi Mat. Nauk'' , '''32''' :  4  (1977)  pp. 55–112</TD></TR></table>