# Can there be a limit cycle without a fixed point in 3D space?

I am working with a population dynamics model. Basically, I have a nonlinear ODE in $R^3$ space, (X,Y,Z), and I know that if I start in the an open region ($0<X<1,0<Y<1,0<Z<1$, basically each species with positive density), then it will stay in that region. I can prove that (0,0,0) is a stable fixed point, and that (X*,Y*,0) is an unstable fixed point. I can also prove that there are no fixed points (stable nor unstable) when all species are present.

Does the lack of fixed points in the bounded area mean that there can't be a limit cycle? I know it would in 2 dimensions, but I don't know if that rule generalizes to 3 dimensions. And I know that I cannot rule out chaos, though that doesn't worry me.

Thank you.

## 1 Answer

If you are just to find a model with a limit cycle but no fixed point in $\mathbb{R}^3$, the answer is ''yes''. As shown in the Figs, the example exists. But if you want the exact equations of the model, it will take much time to construct them.

• Had almost the same example in mind :) the equation would be easier to derive using cylindrical coordinate system. Commented Jun 17, 2016 at 5:37
• Yes, you are right. But it will still cost some time to do so. Usually I just draw a figure to verify the possibility. It is much faster. Commented Jun 17, 2016 at 7:05
• Could you explain more on the figures? I could not understand it. Commented Nov 13, 2019 at 18:30