View source for Lyapunov stability

''of a point relative to a family of mappings

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of a certain space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l0611602.png" />''

Equicontinuity of this family of mappings at the point (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l0611603.png" /> is the set of non-negative numbers in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l0611604.png" />; for example, the real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l0611605.png" /> or the integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l0611606.png" />). Lyapunov stability of a point relative to the family of mappings

is equivalent to the continuity at this point of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l0611607.png" /> of a neighbourhood of this point into the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l0611608.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l0611609.png" />, equipped with the topology of uniform convergence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116010.png" />. Lyapunov stability of a point relative to a mapping is defined as Lyapunov stability relative to the family of non-negative powers of this mapping. Lyapunov stability of a point relative to a dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116011.png" /> is Lyapunov stability of this point relative to the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116012.png" />. Lyapunov stability of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116013.png" /> of an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116014.png" /> given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116015.png" /> is Lyapunov stability of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116016.png" /> relative to the family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116017.png" />.

Lyapunov stability of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116018.png" /> of a differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116019.png" /> given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116020.png" /> is Lyapunov stability of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116021.png" /> relative to the family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116023.png" /> is the [[Cauchy operator|Cauchy operator]] of this equation. Lyapunov stability of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116024.png" /> of a differential equation

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of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116026.png" />, given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116027.png" />, is Lyapunov stability of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116028.png" /> of the corresponding first-order differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116029.png" />, given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116030.png" />, where

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The definitions 1–7 given below are some concrete instances of the above and related definitions.

1) Let a differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116033.png" /> be given, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116034.png" /> lies in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116035.png" />-dimensional normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116036.png" />. A solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116037.png" /> of this equation is called Lyapunov stable if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116038.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116039.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116040.png" /> satisfying the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116041.png" />, the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116042.png" /> of the Cauchy problem

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is unique, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116044.png" />, and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116045.png" /> satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116046.png" />. If, in addition, one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116047.png" /> such that for every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116048.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116049.png" /> whose initial value satisfies the inequality

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the equation

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holds (respectively, the inequality

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holds; here and elsewhere one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116053.png" />), then the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116054.png" /> is called asymptotically (respectively, exponentially) stable.

A solution of the equation

<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/l061160/l06116055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116056.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116057.png" />, is called Lyapunov stable (asymptotically, exponentially stable) if it becomes such after equipping the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116058.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116059.png" />) with a norm. This property of the solution does not depend on the choice of the norm.

2) Let a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116060.png" /> be given, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116061.png" /> is a metric space. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116062.png" /> is called Lyapunov stable relative to the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116063.png" /> if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116064.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116065.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116066.png" /> satisfying the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116067.png" />, the inequality

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holds for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116069.png" />. If, moreover, one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116070.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116071.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116072.png" /> one has the equation

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(the inequality

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respectively), then the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116075.png" /> is called asymptotically (respectively, exponentially) stable relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116076.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116077.png" /> be a mapping from a compact topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116078.png" /> into itself. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116079.png" /> is called Lyapunov stable (asymptotically stable) relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116080.png" /> if it becomes such after equipping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116081.png" /> with a metric. This property of the point does not depend on the choice of the metric.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116082.png" /> is a compact differentiable manifold, then a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116083.png" /> is called exponentially stable relative to a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116084.png" /> if it becomes such after equipping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116085.png" /> with a certain Riemannian metric. This property of the point does not depend on the choice of the Riemannian metric.

3) Suppose that a differential equation (2) is given, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116086.png" /> lies in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116087.png" />. A solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116088.png" /> of this equation is called Lyapunov stable if for each neighbourhood of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116089.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116091.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116092.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116093.png" /> the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116094.png" /> of the Cauchy problem (2), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116095.png" />, is unique, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116096.png" /> and satisfies the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116097.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116098.png" />. If, in addition, one can find a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l06116099.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160100.png" /> such that for every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160101.png" /> of (2) satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160102.png" /> one has the equation

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(respectively,

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for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160105.png" />), then the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160106.png" /> is called asymptotically (respectively, exponentially) stable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160107.png" /> is a normed space, then this definition may be formulated as in 1 above, if as norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160108.png" /> one takes any norm compatible with the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160109.png" />.

4) Let a differential equation (2) be given on a Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160110.png" /> (for which a Euclidean or a Hilbert space can serve as a model) or, in a more general situation, on a Finsler manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160111.png" /> (for which a normed space can serve as a model); the distance function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160112.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160113.png" />. A solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160114.png" /> of this equation is called Lyapunov stable if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160115.png" /> one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160116.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160117.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160118.png" />, the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160119.png" /> of the Cauchy problem (2), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160120.png" />, is unique, defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160121.png" /> and satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160122.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160123.png" />. If, in addition, one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160124.png" /> such that for every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160125.png" /> of (2) whose initial value satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160126.png" /> one has the equation

<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/l061160/l061160127.png" /></td> </tr></table>

(the inequality

<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/l061160/l061160128.png" /></td> </tr></table>

respectively), then the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160129.png" /> is called asymptotically (respectively, exponentially) stable.

Suppose that the differential equation (2) is given on a compact differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160130.png" />. A solution of this equation is called Lyapunov stable (asymptotically, exponentially stable) if it becomes such when the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160131.png" /> is equipped with some Riemannian metric. This property of the solution does not depend on the choice of the Riemannian metric.

5) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160132.png" /> be a [[Uniform space|uniform space]]. Let

<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/l061160/l061160133.png" /></td> </tr></table>

be a mapping defined on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160134.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160135.png" /> is called Lyapunov stable relative to the family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160136.png" /> if for every entourage <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160137.png" /> there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160138.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160139.png" /> such that the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160140.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160141.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160142.png" />, is a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160143.png" />. If, in addition, there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160144.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160145.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160146.png" /> and every entourage <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160147.png" /> one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160148.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160149.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160150.png" />, then the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160151.png" /> is called asymptotically stable.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160152.png" /> is a compact topological space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160153.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160154.png" />, is a mapping given on some open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160155.png" />, then the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160156.png" /> is called Lyapunov stable (asymptotically stable) relative to the family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160157.png" /> if it becomes such after the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160158.png" /> is equipped with the unique uniform structure that is compatible with the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160159.png" />.

6) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160160.png" /> be a topological space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160161.png" /> an open subspace in it. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160162.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160163.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160164.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160165.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160166.png" />, be a mapping having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160167.png" /> as fixed point. The fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160168.png" /> is called Lyapunov stable relative to the family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160169.png" /> if for every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160170.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160171.png" /> there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160172.png" /> of the same point such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160173.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160174.png" />. If, in addition, there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160175.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160176.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160177.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160178.png" />, then the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160179.png" /> is called asymptotically stable relative to the family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160180.png" />.

7) Lyapunov stability (asymptotic, exponential stability) of a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160181.png" /> of an equation of arbitrary order, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160182.png" />, is understood to mean Lyapunov stability (respectively asymptotic, exponential stability) of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160183.png" /> of the corresponding first-order equation (2), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160184.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160185.png" />.

Definitions 1, 2, 4, 6, 7 include stable motions of systems with a finite number of degrees of freedom (where the equations on manifolds arise naturally when considering mechanical systems with a constraint). Definitions 2–7 include stable motions in the mechanics of continuous media and in other parts of physics, stable solutions of operator equations, functional-differential equations (in particular, equations with retarded arguments) and other equations.

==Study of the stability of an equilibrium position of an autonomous system.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160186.png" /> be an autonomous differential equation defined in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160187.png" />, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160188.png" /> is continuously differentiable and vanishes at this point. If the real parts of all eigen values of the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160189.png" /> are negative, then the fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160190.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160191.png" /> is exponentially stable (Lyapunov's theorem on stability in a first approximation); to facilitate the verification of the condition in this theorem one applies criteria for stability. If under these conditions at least one of the eigen values of the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160192.png" /> has positive real part (this condition may be checked without finding the eigen values themselves, cf. [[Stability criterion|Stability criterion]]), then the fixed point of the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160193.png" /> is unstable.

Example. The equation of the oscillation of a pendulum with friction is

<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/l061160/l061160194.png" /></td> </tr></table>

The lower equilibrium position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160195.png" /> is exponentially stable, since the roots of the characteristic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160196.png" /> of the variational equation (cf. [[Variational equations|Variational equations]]) have negative real parts. The upper equilibrium position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160197.png" /> is unstable, since the characteristic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160198.png" /> of the variational equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160199.png" /> has a positive root. This instability takes place even in the absence of friction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160200.png" />. The lower equilibrium position of a pendulum without friction is one of the so-called critical cases, when all eigen values of the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160201.png" /> are contained in the left complex half-plane, and at least one of them lies on the imaginary axis.

For the study of stability in critical cases, A.M. Lyapunov proposed the so-called second method for studying stability (cf. [[Lyapunov function|Lyapunov function]]). For a pendulum without friction,

<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/l061160/l061160202.png" /></td> </tr></table>

the lower equilibrium position is Lyapunov stable, since there exists a [[Lyapunov function|Lyapunov function]]

<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/l061160/l061160203.png" /></td> </tr></table>

— the total energy of the pendulum; the condition of non-positivity for the derivative of this function is a consequence of the law of conservation of energy.

A fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160204.png" /> of a differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160205.png" /> is exponentially stable relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160206.png" /> if all eigen values of the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160207.png" /> are less than 1 in modulus, and it is unstable if at least one of them has modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160208.png" />.

The study of the stability of periodic points of differentiable mappings reduces to the study of stability of fixed points relative to the powers of these mappings. Periodic solutions of autonomous differential equations are not asymptotically stable (cf. [[Orbit stability|Orbit stability]]; [[Andronov–Witt theorem|Andronov–Witt theorem]]).

It should not be believed that exponential stability of the null solution of the variational equation of the autonomous differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160209.png" /> along a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160210.png" /> implies stability of the solution. This is shown by Perron's example (cf. [[#References|[2]]], [[#References|[3]]]):

<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/l061160/l061160211.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160212.png" /> the null solution of the system of variational 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/l061160/l061160213.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

of the system (3) (along the null solution) is exponentially stable (the Lyapunov characteristic exponents of the system (4) are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160214.png" />, cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]), but for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160215.png" /> the null solution of the system (3) is unstable. However, stability in the first approximation is typical, in a sense explained below.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160216.png" /> be the set of diffeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160217.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160218.png" /> onto itself having uniformly continuous derivatives that satisfy the inequality

<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/l061160/l061160219.png" /></td> </tr></table>

For each diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160220.png" /> denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160221.png" /> the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160222.png" /> satisfying the inequality

<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/l061160/l061160223.png" /></td> </tr></table>

one endows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160224.png" /> with the distance function

<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/l061160/l061160225.png" /></td> </tr></table>

For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160226.png" /> there is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160227.png" /> an everywhere-dense set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160228.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160229.png" /> with the following property: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160230.png" /> is such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160231.png" /> the inequality

<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/l061160/l061160232.png" /></td> </tr></table>

holds, then there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160233.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160234.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160235.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160236.png" /> the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160237.png" /> is exponentially stable relative to the diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160238.png" />.

For a dynamical system given on a compact differentiable manifold, an analogous theorem can be formulated more simply and as a differential-topologically invariant statement. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160239.png" /> be a closed differentiable manifold. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160240.png" /> of all diffeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160241.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160242.png" /> mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160243.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160244.png" /> can be equipped with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160245.png" />-topology. In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160246.png" /> there is an everywhere-dense set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160247.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160248.png" /> with the following property: If for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160249.png" /> the inequality

<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/l061160/l061160250.png" /></td> </tr></table>

holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160251.png" />, then there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160252.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160253.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160254.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160255.png" /> the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160256.png" /> is exponentially stable relative to the diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061160/l061160257.png" />.

The concepts of Lyapunov stability, asymptotic stability and exponential stability were introduced by Lyapunov [[#References|[1]]] in order to develop methods for studying stability in the sense of these definitions (cf. [[Lyapunov stability theory|Lyapunov stability theory]]).

====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">  O. Perron,   "Ueber Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen"  ''Math. Z.'' , '''29'''  (1928)  pp. 129–160</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.E. Bellman,   "Stability theory of differential equations" , Dover, reprint  (1969)</TD></TR></table>



====Comments====
For stability questions of differential equations with discontinuous right-hand sides cf. [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.P. Lasalle,   S. Lefschetz,   "Stability by Lyapunov's direct method with applications" , Acad. Press  (1961)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.W. Hirsch,   S. Smale,   "Differential equations, dynamical systems, and linear algebra" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Hahn,   "Stability of motion" , Springer  (1967)  pp. 422</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N.P. Bhatia,   G.P. Szegö,   "Stability theory of dynamical systems" , Springer  (1970)  pp. 30–36</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.I. Rabinovich,   D.I. Trubetskov,   "Oscillations and waves in linear and nonlinear systems" , Kluwer  (1989)  pp. Chapt. 6–7  (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.F. Filippov,   "Differential equations with discontinuous righthand sides" , Kluwer  (1989)  pp. §15  (Translated from Russian)</TD></TR></table>