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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c0220601.png" />, defined in a neighbourhood of a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c0220602.png" /> of a system of ordinary differential equations
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{{TEX|done}}
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A function $v(x)$, defined in a neighbourhood of a fixed point $x=0$ of a system of ordinary differential 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/c/c022/c022060/c0220603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\dot x=F(x),\quad x\in\mathbf R^n,\quad F(0)=0,\tag{*}$$
  
and satisfying the two conditions: 1) there exists a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c0220604.png" /> with the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c0220605.png" /> on its boundary in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c0220606.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c0220607.png" /> on the boundary of the domain close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c0220608.png" />; and 2) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c0220609.png" /> the derivative along the flow of the system (*) (cf. [[Differentiation along the flow of a dynamical system|Differentiation along the flow of a dynamical system]]) satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c02206010.png" />.
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and satisfying the two conditions: 1) there exists a domain $G$ with the point $x=0$ on its boundary in which $v>0$, and $v=0$ on the boundary of the domain close to $x=0$; and 2) in $G$ the derivative along the flow of the system \ref{*} (cf. [[Differentiation along the flow of a dynamical system|Differentiation along the flow of a dynamical system]]) satisfies $\dot v>0$.
  
Chetaev's theorem [[#References|[1]]] holds: If there is a Chetaev function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c02206011.png" /> for the system (*), then the fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c02206012.png" /> is Lyapunov unstable.
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Chetaev's theorem [[#References|[1]]] holds: If there is a Chetaev function $v$ for the system \ref{*}, then the fixed point $x=0$ is Lyapunov unstable.
  
 
A Chetaev function is a generalization of a [[Lyapunov function|Lyapunov function]] and gives a convenient way of proving instability (cf. [[#References|[2]]]). For example, for the system
 
A Chetaev function is a generalization of a [[Lyapunov function|Lyapunov function]] and gives a convenient way of proving instability (cf. [[#References|[2]]]). For example, for the system
  
<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/c/c022/c022060/c02206013.png" /></td> </tr></table>
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$$\dot x=ax+o(|x|+|y|),$$
  
<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/c/c022/c022060/c02206014.png" /></td> </tr></table>
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$$\dot y=-by+o(|x|+|y|),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c02206015.png" />, a Chetaev function is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c02206016.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022060/c02206017.png" />. Generalizations of Chetaev functions have been suggested, in particular for non-autonomous systems (cf. [[#References|[3]]]).
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where $a,b>0$, a Chetaev function is $v=x^2-c^2y^2$ for any $c\neq0$. Generalizations of Chetaev functions have been suggested, in particular for non-autonomous systems (cf. [[#References|[3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.G. Chetaev,  "A theorem on instability"  ''Dokl. Akad. Nauk SSSR'' , '''1''' :  9  (1934)  pp. 529–531  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.G. Chetaev,  "Stability of motion" , Moscow  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Krasovskii,  "Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay" , Stanford Univ. Press  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Rouche,  P. Habets,  M. Laloy,  "Stability theory by Liapunov's direct method" , Springer  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.G. Chetaev,  "A theorem on instability"  ''Dokl. Akad. Nauk SSSR'' , '''1''' :  9  (1934)  pp. 529–531  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.G. Chetaev,  "Stability of motion" , Moscow  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Krasovskii,  "Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay" , Stanford Univ. Press  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Rouche,  P. Habets,  M. Laloy,  "Stability theory by Liapunov's direct method" , Springer  (1977)</TD></TR></table>

Revision as of 10:06, 19 August 2014

A function $v(x)$, defined in a neighbourhood of a fixed point $x=0$ of a system of ordinary differential equations

$$\dot x=F(x),\quad x\in\mathbf R^n,\quad F(0)=0,\tag{*}$$

and satisfying the two conditions: 1) there exists a domain $G$ with the point $x=0$ on its boundary in which $v>0$, and $v=0$ on the boundary of the domain close to $x=0$; and 2) in $G$ the derivative along the flow of the system \ref{*} (cf. Differentiation along the flow of a dynamical system) satisfies $\dot v>0$.

Chetaev's theorem [1] holds: If there is a Chetaev function $v$ for the system \ref{*}, then the fixed point $x=0$ is Lyapunov unstable.

A Chetaev function is a generalization of a Lyapunov function and gives a convenient way of proving instability (cf. [2]). For example, for the system

$$\dot x=ax+o(|x|+|y|),$$

$$\dot y=-by+o(|x|+|y|),$$

where $a,b>0$, a Chetaev function is $v=x^2-c^2y^2$ for any $c\neq0$. Generalizations of Chetaev functions have been suggested, in particular for non-autonomous systems (cf. [3]).

References

[1] N.G. Chetaev, "A theorem on instability" Dokl. Akad. Nauk SSSR , 1 : 9 (1934) pp. 529–531 (In Russian)
[2] N.G. Chetaev, "Stability of motion" , Moscow (1965) (In Russian)
[3] N.N. Krasovskii, "Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay" , Stanford Univ. Press (1963) (Translated from Russian)
[4] N. Rouche, P. Habets, M. Laloy, "Stability theory by Liapunov's direct method" , Springer (1977)
How to Cite This Entry:
Chetaev function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chetaev_function&oldid=32994
This article was adapted from an original article by A.D. Bryuno (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article