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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
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A theorem that permits one to establish the absence of closed trajectories of [[dynamical system]]s in the plane, defined by the 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/b/b015/b015530/b0155301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$x'=P(x,y),\quad y'=Q(x,y).\label{*}\tag{*}$$
  
The criterion was first formulated by I. Bendixson [[#References|[1]]] as follows: If in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b0155302.png" /> the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b0155303.png" /> has constant sign (i.e. the sign remains unchanged and the expression vanishes only at isolated points or on a curve), then the system (*) has no closed trajectories in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b0155304.png" />. This criterion was generalized by H. Dulac [[#References|[2]]] as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b0155305.png" /> is a simply-connected domain in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b0155306.png" />-plane, if the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b0155307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b0155308.png" />, and if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b0155309.png" /> can be found such that
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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 \eqref{*} 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
  
<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/b/b015/b015530/b01553010.png" /></td> </tr></table>
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$$\int\int\limits_D\left\lbrace\frac{\partial(fP)}{\partial x}+\frac{\partial(fQ)}{\partial y}\right\rbrace dxdy\neq0$$
  
for any simply-connected subdomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b01553011.png" />, then the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b01553012.png" /> does not contain any simple rectifiable closed curve consisting of trajectories and singular points of the system (*). If the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015530/b01553013.png" /> is an annulus, a similar theorem states that a closed trajectory of (*), if it exists, is unique. A generalization applying to the case of system (*) with cylindrical phase space [[#References|[3]]] is also possible.
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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 [[#References|[3]]] is also possible.
  
 
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<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>
 
<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>
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[[Category:Dynamical systems and ergodic theory]]

Latest revision as of 15:57, 14 February 2020

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 [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 \eqref{*} has no closed trajectories in the domain $G$. This criterion was generalized by H. Dulac [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 \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 [3] is also possible.

References

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


Comments

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

References

[a1] A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)
How to Cite This Entry:
Bendixson criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bendixson_criterion&oldid=15963
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