I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a worked example and my doubts.
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
$$x'=P(x,y),\quad y'=Q(x,y).\label{*}\tag{*}$$. According to Bendixson criterion, if in a simply-connected domain $G$ the expression
$$I(x,y)=P_x'+Q_y'\label{**}\tag{**}$$ has constant sign, then the system \eqref{*} has no closed trajectories in the domain $G$. In other words, if the expression $\eqref{**}$ 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
$$x'=-y+x(x^{2}+y^{2}-1),\quad y'=x+y(x^{2}+y^{2}-1).\label{***}\tag{***}$$ 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.~(\ref{**}), 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:
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?
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
- 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+%2Bnon-simply+connected+regions&source=bl&ots=01UFdC1xHN&sig=ACfU3U2AITbcvXiP0mi8Lf8BJ89vRmYN9g&hl=pt-BR&sa=X&ved=2ahUKEwic3NCvroz2AhV-K7kGHTVTD8IQ6AF6BAgsEAM#v=onepage&q&f=false)