I Limit cycles, differential equations and Bendixson's criterion

[Vini]

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

 

[mathwonk]

have you read a definition of simply connected? to a topologist, it means that every simple closed curve can be deformed continuously to a point within the region. To a complex analyst it is sometimes taken to mean that the complement of the region, in the Riemann sphere, i.e. the one point compactification of the plane, is connected. This second version can be used in your case to see quickly that the interior of a circle is simply connected, but the exterior, in the plane, is not. I.e. the complement in the sphere, of the interior of a disc, is itself a disc, but the complement in the sphere, of the (plane) exterior of a disc, is both the interior of that disc and the point at infinity.