Conditional stability
of a point relative to a family of mappings
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Equicontinuity at this point of the family of restrictions of the mappings
to a certain manifold
imbedded in
(with the induced metric on
); here
is the set of real or integer non-negative numbers:
or
.
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 is the conditional stability of this point relative to the family of mappings
. The conditional stability of a solution of an equation
![]() |
given on is the conditional stability of the point
relative to the family of mappings
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The conditional stability of the solution of a differential equation
![]() | (2) |
given on is the conditional stability of the point
relative to the family of mappings
, where
is the Cauchy operator of this equation. The conditional stability of the solution
of a differential equation of order
,
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given on , is the conditional stability of the solution
, given on
, of the corresponding first-order differential equation of the form (2), where
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The definitions 1)–5) below are some concrete examples of these and related notions.
1) Given a differential equation (2), where is a normed
-dimensional vector space and
. The solution
of this equation is called conditionally stable with index
if there is a
-dimensional disc
imbedded in
(considered as a manifold of class
), containing the point
and having the following property: For each
there is a
such that for every
satisfying the inequality
, the solution
of the same equation satisfying the initial condition
is uniquely defined on
, and for each
satisfies the inequality
. If the disc
with the given property may be chosen so that
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(respectively,
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here, and elsewhere is understood that ) for every solution of the same equations starting in this disc (i.e. such that
), then the solution
is called asymptotically (respectively, exponentially) conditionally stable (with index
).
The solution of the equation (2) ( or
) is called conditionally (asymptotically, exponentially conditionally) stable with index
, if it becomes such as the result of equipping
(or
) with a suitable norm. This property of the solution does not depend on the choice of norm.
2) Given an -dimensional Riemannian manifold
(the distance function on which is written as
), one calls a point
conditionally stable (with index
) relative to a mapping
if there is a (usually smooth)
-dimensional disc
imbedded in
, containing the point
and having the following property: For each
there exists a
such that for every
satisfying the inequality
, the inequality
holds for all
. If the disc
with the above property can be chosen so that
![]() |
(respectively,
![]() |
for each , then the point
is called asymptotically (respectively, exponentially) conditionally stable (with index
) relative to the mapping
.
Let be a compact differentiable manifold. A point
is called conditionally stable (asymptotically, exponentially conditionally stable) with index
relative to a mapping
if it becomes such as a result of equipping
with a suitable Riemannian metric. This property of
does not depend on the choice of the Riemannian metric on
.
3) Consider the differential equation (2) on an -dimensional Riemannian (or Finsler, cf. Finsler geometry) manifold
, the distance function on which is denoted by
. The solution
of this equation is called conditionally stable (with index
) if there is a
-dimensional disc
imbedded in
(considered as a manifold of class
, where usually
), containing the point
and having the following property: For each
there exists a
such that for every
satisfying the inequality
, the solution
of the same equation satisfying the initial condition
is unique, defined on
, and for each
satisfies the inequality
. If the disc
with the above property may be taken so that
![]() |
(respectively,
![]() |
for every solution of the same equation starting in this disc (i.e. such that ), then the solution
is called asymptotically (respectively, exponentially) conditionally stable (with index
).
4) Let be an
-dimensional manifold of class
and let
be an open subset of it. Suppose that a point
is fixed under a family of mappings
of class
(
, where
is
or
). The fixed point
is called conditionally stable (with index
) relative to the family of mappings
if there is
-dimensional disc
smoothly imbedded (by an imbedding of class
) in
such that for every neighbourhood
of
there is a neighbourhood
of the same point such that
for every
. If the disc
with this property may be taken so that
for every
, then the fixed point
is called asymptotically conditionally stable (with index
) relative to the family of mappings
.
5) The conditional (conditional asymptotic, conditional exponential) stability (with index ) of the solution
of an equation of arbitrary order
, is defined as the conditional (asymptotic, conditional exponential) stability (with index
) of the solution
of the corresponding first-order equation (2), where
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Sometimes (cf. e.g. [3]) in defining conditional stability one requires the index to be non-zero: conditional stability with index zero always holds. Conditional stability (conditional asymptotic, conditional exponential stability) with index
(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 an autonomous differential equation
![]() | (3) |
is given, the right-hand side of which is continuously differentiable and vanishes at the point . If in the open left half-plane in the complex plane there are
eigen values of the derivative
, then this fixed point of equation (3) is conditionally exponentially stable with index
(Lyapunov's theorem on conditional stability). For example, the upper equilibrium position
,
of the equation of oscillation of a pendulum
is exponentially conditionally stable with index 1, because one of the roots of the characteristic equation
of the variational equation (cf. Variational equations)
is negative.
A fixed point of a differentiable mapping
is exponentially conditionally stable with index
relative to
if
eigen values of the derivative
lie in the open unit disc. A periodic point
of a differential mapping
, having period
, is conditionally (asymptotically conditionally, exponentially conditionally) stable with index
relative to
if and only if it has this property relative to
.
A periodic solution of an autonomous differential equation (3) with smooth right-hand side having period
is (asymptotically, exponentially) conditionally stable with index
if and only if its value at the point
is (respectively, asymptotically, exponentially) conditionally stable with index
relative to the mapping
, where
is the Cauchy operator of (3).
The example of O. Perron (cf. Lyapunov stability) shows that the negativity of Lyapunov exponents of the variational equation along the solution of (3) does not imply the conditional stability with index
of this solution. However, one has the following theorem, which shows that the situation described be Perron's example is not generic.
1) Let be the set of all diffeomorphisms
of a Euclidean space
having uniformly continuous derivatives satisfying the inequality
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For every diffeomorphism denote by
the set of diffeomorphisms
satisfying the inequality
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on the set the distance function
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is given.
Fir each one has in
an everywhere-dense set
of type
with the following property: For every
the point
is exponentially conditionally stable relative to the diffeomorphism
with index
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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 be a closed differentiable manifold. The set
of all diffeomorphisms
of class
mapping
onto
is equipped with the
-topology. In the space
there is an everywhere-dense set
of type
with the following property: For each
the point
is exponentially conditionally stable relative to the diffeomorphism
with index
![]() | (4) |
3) For every diffeomorphism of a closed differentiable manifold
and for every probability distribution on
that is invariant relative to
(and the
-algebra of which contains all Borel sets), the set of points
that are exponentially conditionally stable with index (4) relative to
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 |
Conditional stability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_stability&oldid=46444