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Difference between revisions of "Stability region"

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A set in the space of values of the parameter on which the [[Cauchy problem|Cauchy problem]] depends. This set (which is not a domain, in general) is the union of the connected components of the interior of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087040/s0870401.png" /> of values of the parameter for which the solution of the Cauchy problem is Lyapunov stable (cf. [[Lyapunov stability|Lyapunov stability]]) and the set of all points on the boundary of these components which belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087040/s0870402.png" />. The definition quoted is an attempt to give a precise meaning to a concept usually described in a more or less diffuse manner (cf. [[#References|[1]]], p. 194, 195, 197).
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A set in the space of values of the parameter on which the [[Cauchy problem|Cauchy problem]] depends. This set (which is not a domain, in general) is the union of the connected components of the interior of the set $S$ of values of the parameter for which the solution of the Cauchy problem is Lyapunov stable (cf. [[Lyapunov stability|Lyapunov stability]]) and the set of all points on the boundary of these components which belong to $S$. The definition quoted is an attempt to give a precise meaning to a concept usually described in a more or less diffuse manner (cf. [[#References|[1]]], p. 194, 195, 197).
  
 
Example. The null solution of the [[Mathieu equation|Mathieu equation]]
 
Example. The null solution of the [[Mathieu equation|Mathieu 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/s/s087/s087040/s0870403.png" /></td> </tr></table>
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$$\ddot y+(\delta+\epsilon\cos t)y=0,$$
  
depending on the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087040/s0870404.png" />, has a countable set of stability regions (cf. [[#References|[1]]], Figure 78). Among these are regions meeting the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087040/s0870405.png" />, which explains the possibility of stabilization of the upper position of equilibrium of a pendulum by means of periodic (sinusoidal) oscillations of the point of the hanger in the vertical direction (cf. [[#References|[1]]], Chapt. VI, Sect. 1.4).
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depending on the parameters $(\delta,\epsilon)\in\mathbf R^2$, has a countable set of stability regions (cf. [[#References|[1]]], Figure 78). Among these are regions meeting the half-plane $\delta<0$, which explains the possibility of stabilization of the upper position of equilibrium of a pendulum by means of periodic (sinusoidal) oscillations of the point of the hanger in the vertical direction (cf. [[#References|[1]]], Chapt. VI, Sect. 1.4).
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
For a difference method used to solve an ordinary differential equation numerically, discrete analogues of this region exist. In particular, such a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087040/s0870406.png" /> then refers to a part of the complex plane; often a model problem like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087040/s0870407.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087040/s0870408.png" /> complex) is used for this.
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For a difference method used to solve an ordinary differential equation numerically, discrete analogues of this region exist. In particular, such a set $S$ then refers to a part of the complex plane; often a model problem like $\dot x=\lambda x$ ($\lambda$ complex) is used for this.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Harter,  S. Nörgett,  G. Wanner,  "Solving ordinary differential equations" , '''I''' , Springer  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Harter,  S. Nörgett,  G. Wanner,  "Solving ordinary differential equations" , '''I''' , Springer  (1987)</TD></TR></table>

Latest revision as of 16:58, 15 September 2014

A set in the space of values of the parameter on which the Cauchy problem depends. This set (which is not a domain, in general) is the union of the connected components of the interior of the set $S$ of values of the parameter for which the solution of the Cauchy problem is Lyapunov stable (cf. Lyapunov stability) and the set of all points on the boundary of these components which belong to $S$. The definition quoted is an attempt to give a precise meaning to a concept usually described in a more or less diffuse manner (cf. [1], p. 194, 195, 197).

Example. The null solution of the Mathieu equation

$$\ddot y+(\delta+\epsilon\cos t)y=0,$$

depending on the parameters $(\delta,\epsilon)\in\mathbf R^2$, has a countable set of stability regions (cf. [1], Figure 78). Among these are regions meeting the half-plane $\delta<0$, which explains the possibility of stabilization of the upper position of equilibrium of a pendulum by means of periodic (sinusoidal) oscillations of the point of the hanger in the vertical direction (cf. [1], Chapt. VI, Sect. 1.4).

References

[1] J.J. Stoker, "Nonlinear vibrations in mechanical and electrical systems" , Interscience (1950)
[2] N.N. Bautin, "The behaviour of dynamical systems near the border of their region of stability" , Moscow (1984) (In Russian)


Comments

For a difference method used to solve an ordinary differential equation numerically, discrete analogues of this region exist. In particular, such a set $S$ then refers to a part of the complex plane; often a model problem like $\dot x=\lambda x$ ($\lambda$ complex) is used for this.

References

[a1] E. Harter, S. Nörgett, G. Wanner, "Solving ordinary differential equations" , I , Springer (1987)
How to Cite This Entry:
Stability region. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stability_region&oldid=33301
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article