I took a look at the paper of Hastings and Murray. The proof is rather cute. I guess the monotonicity is also manifest in their proof of the cycling condition among the sub-boxes.

I don’t have too much background in dynamical systems, so another quick question: let us put ourselves in the shoes of Hastings and Murray. Suppose we have a three dimensional dynamical system. We know that it is monotone. We also know that there exists a convex set with smooth boundary (say that’s true. I know they have something slightly weaker) such that two things holds.

(a) The flow on the boundary points inwards. (By Brouwer’s this implies existence of at least one fixed point.)
(b) The fixed point is unique and unstable.

Then is it sufficient to just apply Poincare-Benedixson and say that “if we can show there are no homoclinic trajectories then there must exist an orbit”? Or does that actually require more work. (I am just wondering if there are more abstract/general ways of arriving at Hastings and Murray’s result. Another way of phrasing my question is probably “is it s theorem that Poincare-Benedixson is applicable to three-dimensional monotone systems, or is there a caveat?”)

Thanks in advance for your time.

That said, I saw the BZ reaction in a chemistry class once. After primed by a semester of The Second Law and Thermodynamic Equilibrium (in a different class, but still fresh on the memory), that was the most counter-intuitive thing.