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Difference between revisions of "Chetaev function"

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(Category:Ordinary differential equations)
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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
 
A function $v(x)$, defined in a neighbourhood of a fixed point $x=0$ of a system of ordinary differential equations
  
$$\dot x=F(x),\quad x\in\mathbf R^n,\quad F(0)=0,\tag{*}$$
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$$\dot x=F(x),\quad x\in\mathbf R^n,\quad F(0)=0,\label{*}\tag{*}$$
  
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|Differentiation along the flow of a dynamical system]]) satisfies $\dot v>0$.
+
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 \eqref{*} (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 $v$ for the system \ref{*}, then the fixed point $x=0$ is Lyapunov unstable.
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Chetaev's theorem [[#References|[1]]] holds: If there is a Chetaev function $v$ for the system \eqref{*}, 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

Latest revision as of 15:31, 14 February 2020

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

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

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 \eqref{*} (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 \eqref{*}, 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=44703
This article was adapted from an original article by A.D. Bryuno (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article