View source for Bendixson criterion

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

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

The criterion was first formulated by I. Bendixson [[#References|[1]]] 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 \ref{*} has no closed trajectories in the domain $G$. This criterion was generalized by H. Dulac [[#References|[2]]] 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 \ref{*}. If the domain $G$ is an annulus, a similar theorem states that a closed trajectory of \ref{*}, if it exists, is unique. A generalization applying to the case of system \ref{*} with cylindrical phase space [[#References|[3]]] is also possible.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Bendixson,   "Sur les courbes définies par des équations différentielles"  ''Acta Math.'' , '''24'''  (1901)  pp. 1–88</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Dulac,   "Récherches des cycles limites"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''204'''  (1937)  pp. 1703–1706</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Andronov,   A.A. Vitt,   A.E. Khaikin,   "Theory of oscillators" , Pergamon  (1966)  (Translated from Russian)</TD></TR></table>



====Comments====
Bendixson's criterion is also called the Poincaré–Bendixson theorem.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Andronov,   E.A. Leontovich,   I.I. Gordon,   A.G. Maier,   "Qualitative theory of second-order dynamic systems" , Wiley  (1973)  (Translated from Russian)</TD></TR></table>

[[Category:Dynamical systems and ergodic theory]]