Periodic orbits enclosing fixed points in a differential equation

Question

Suppose we have a differential equation $\mathbf{\dot{x}}=\mathbf{f(x)}$ for $\mathbf{x}\in \Bbb{R}^2$.

Suppose further that there are three fixed points, of which one is a saddle and two are sinks. I am not sure how to determine examples of the following scenarios or to prove they do not exist:

1. There exists a periodic orbit enclosing precisely one sink.
2. There exists a periodic orbit enclosing all three fixed points.

The index test does not rule out either of these possibilities and I am unsure how to construct examples demonstrating existence.

Any help would be much appreciated!

Answer 1

As you do not have any specific question or attempt to build up upon, here is some general strategy:

The most important insight for this kind of exercise is that you can do almost everything you want once you leave the direct vicinity of the attractors. You can just glue together piecewise dynamical systems, modify the inside of a limit cycle, etc. In this case, you have to take care of not introducing additional fixed points, but that’s it.

As a consequence, whenever you can draw a reasonably complete phase-space diagram of such a system, an answer exists. It may be tedious to construct actual ODEs that fulfil it, but depending on whom you need to convince, you do not even need to jump through that hoop. Take care that all the homo- and heteroclinic orbits end somewhere (and be it infinity) and the continuity of the phase-space flow becomes clear (e.g., no adjacent trajectories running in opposite direction).

With that being said, some specific hints:

• Note how the Problem 1 can be reduced to finding of finding a periodic orbit enclosing one sink. The presence of the other two fixed points somewhere in phase does not affect it.

• Note that when you found a solution to Problem 1, you can solve Problem 2 by finding a configuration that behaves like a sink to the outside (all trajectories ingoing), but has the three fixed points in question on the inside.

• Look at some bifurcations. Is there a bifurcation which features a sink within a limit cycle? Is there a bifurcation which turns a sink into two sinks and one saddle?