# 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
How to Cite This Entry:
Conditional stability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_stability&oldid=46514
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article