Conditional stability

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 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) 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. [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 (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) $ \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) 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).

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

[1]	A.M. Lyapunov, "Collected works" , 2 , Moscow-Leningrad (1956) (In 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]	B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow (1967) (In Russian)
[4]	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
[5]	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