# Bendixson criterion

A theorem that permits one to establish the absence of closed trajectories of dynamical systems in the plane, defined by the equation

$$x'=P(x,y),\quad y'=Q(x,y).\label{*}\tag{*}$$

The criterion was first formulated by I. Bendixson  as follows: If in a simply-connected domain $G$ the expression $P_x'+Q_y'$ has constant sign (i.e. the sign remains unchanged and the expression vanishes only at isolated points or on a curve), then the system \eqref{*} has no closed trajectories in the domain $G$. This criterion was generalized by H. Dulac  as follows: If $G$ is a simply-connected domain in the $(x,y)$-plane, if the functions $P$ and $Q\in C^1(G)$, and if a function $f(x,y)\in C^1(G)$ can be found such that

$$\int\int\limits_D\left\lbrace\frac{\partial(fP)}{\partial x}+\frac{\partial(fQ)}{\partial y}\right\rbrace dxdy\neq0$$

for any simply-connected subdomain $D\subset G$, then the domain $G$ does not contain any simple rectifiable closed curve consisting of trajectories and singular points of the system \eqref{*}. If the domain $G$ is an annulus, a similar theorem states that a closed trajectory of \eqref{*}, if it exists, is unique. A generalization applying to the case of system \eqref{*} with cylindrical phase space  is also possible.

How to Cite This Entry:
Bendixson criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bendixson_criterion&oldid=44727
This article was adapted from an original article by r equation','../w/w097310.htm','Whittaker equation','../w/w097840.htm','Wronskian','../w/w098180.htm')" style="background-color:yellow;">N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article