Topology, limit cycles, Bendixson's criterion, and simply connected regions

Question

I have some doubts about the criterion of Bendixson that regards the non-existence of limit cycles.

According to Bendixson's theorem, let

$$x'=P(x,y),\quad y'=Q(x,y).\label{*}\tag{1}$$ be a two-dimensional system. If in a simply-connected domain $G$, the expression $\partial_{x} P+\partial_{y}Q$ has a constant sign (i.e. the sign remains unchanged and the expression vanishes only at isolated points or on a curve), then the system (1) has no closed trajectories in the domain $G$. That is to say, the theorem allows one to establish the absence of closed trajectories (limit cycles) in the plane for an autonomous dynamical system.

According to traditional logic, one may see that this criterion is a conditional statement. If we analyze the counter-positive form of that statement, then it would allow us to infer that the two-dimensional system may have closed trajectories in non-simply connected regions.

Based on the above, I have three related questions:

1. Is it possible to have limit cycles in non-simply connected regions?

2. From the topological viewpoint, may singularities be seen as holes in the domain G?

3. Could manifolds with singularities be considered simply-connected?

Answer 1

First, a small note: closed trajectory is not the same as a limit cycle. For example, conservative systems have many closed trajectories, but not limit cycles (think of the simple harmonic oscillator). A limit cycle is more specific, it means that in either forwards or backwards time, nearby initial conditions are attracted to the cycle. Now, to deal with your questions:

1. "Is it possible to have limit cycles in non-simply connected regions"? Sure, take your favorite limit cycle oscillator in $\mathbb{R}^2$ for which the origin is an unstable fixed point(for example the Van der Pol oscillator) and remove the origin. The system is still well defined, since no trajectories would have passed through the origin. Now any domain containing the limit cycle is not simply connected.
2. "From the topological viewpoint, may singularities be seen as holes in the domain G?" Not necessarily. For example, every vector field on the sphere $S^2$ has a singularity (see the Hairy Ball Theorem), but the sphere is simply connected, i.e. has no 1-dimensional holes. What is true is that if you have a singular vector field, i.e. a vector field with a fixed point, then the same vector field can be defined on the domain minus the fixed point, since no trajectory will ever reach the fixed point in finite time (forwards or backwards).
3. "Could manifolds with singularities be considered simply connected?" It depends on what you mean by "manifolds with singularities." I assume you mean "manifolds which admit a vector field with singularities" in which case the answer is no. Admitting a vector field with singularities is a distinct concept from being simply connected. Some manifolds are simply connected but only admit vector fields with singularities (again, $S^2$ is an example) and some manifolds are not simply connected, but admit vector fields with singularities (for an example, take $\mathbb{R}^2$ and a vector field with 2 fixed points and just remove one of the fixed points. The resulting space is not simply connected, but admits a vector field with a singularity."

I hope this clarifies some things!