- #1

Vini

- 3

- 1

The Bendixson criterion is a theorem that permits one to establish the absence of closed trajectories of dynamical systems in the plane, defined by the equation

$$x'=P(x,y),\quad y'=Q(x,y) \qquad(*)$$

According to Bendixson criterion, if in a simply-connected domain ##G## the expression

$$I(x,y)=P_x'+Q_y'\qquad(**)$$

has constant sign, then the system(*) has no closed trajectories in the domain ##G##. In other words, if the expression (**) is not identically zero, the sign remains unchanged and the expression vanishes only at isolated points or on a curve, so there are no closed trajectories, either periodic solutions or a singular closed trajectory inside region ##G##.

Having presented the theorem and its interpretation, now I turn to a simple application of it.

As a worked example (see example III of Ref.[1] below), let us consider

$$x'=-y+x(x^{2}+y^{2}-1),\quad y'=x+y(x^{2}+y^{2}-1) \qquad (***)$$

as a dynamical system in two-dimensions. Suppose that we are asked to prove that there no closed trajectories in the region ##G## inside the circle with center at ##(0,0)## and radius ##\displaystyle\frac{1}{\sqrt{2}}##.

Based on the theorem above, let ##P(x,y)=-y+x(x^{2}+y^{2}-1)## and ## Q(x,y)=x+y(x^{2}+y^{2}-1)##. By employing Eq.~(**), one can readily see that

##I(x,y)=4\left(x^{2}+y^{2}-\frac{1}{2}\right)##,

in which is evident that ##I(x,y)## has a constant sign inside and outside the circle ##x^{2}+y^{2}=1/2##. Based on that result, I have a few doubts:

1. According to Ref.[1], the given system has no closed orbits inside the circle which is interpreted as a simply-connected region. Why is the region G inside the circle simply connected?

2. According to the same Ref.[1], the region outside the circle is considered non-simply connected. Why is such region non-simply connected? that is, what property does the region outsided the circle have in order to be defined as non-simply connected?

References

1. Layek, G.C.. An Introduction to Dynamical Systems and Chaos. Índia: Springer India, 2015. Page 174. (See https://books.google.com.br/books?id=wfcUCwAAQBAJ&pg=PA174&lpg=PA174&dq=bendixson+criterion++non-simply+connected+regions&source=bl&ots=01UFdC1xHN&sig=ACfU3U2AITbcvXiP0mi8Lf8BJ89vRmYN9g&hl=pt-BR&sa=X&ved=2ahUKEwic3NCvroz2AhV-K7kGHTVTD8IQ6AF6BAgsEAM#v=onepage&q&f=false)