# Do chaos and/or limit cycles always require the existence of an unstable fixed point?

Consider an arbitrary dynamical system in arbitrary dimension. What is the connection between unstable fixed point and chaos and limit cycles? Does chaos and/or limit cycles require the existence of an unstable fixed point?

I know that this is true in some specific cases. In two dimensions limit cycles have an unstable fixed point in the interior. In the Generalized Lotka-Volterra equations, both chaos and limit cycles require the existence of an unstable fixed point [see Evolutionary Games and Population Dynamics, Hofbauer and Sigmund].

Is this true in a more general setting?

• Incidentally, in two dimensions stable limit cycles have a repellor in the interior, which could be another unstable limit cycle or an unstable fixed point. An unstable limit-cycle could well have a stable fixed point on it's interior (or a stable limit-cycle). Jan 26, 2018 at 7:54

No, it isn't. One of the counterexamples is the Sprott "A" system: $$\left\{\begin{array}{lll} \dot x&=&y\\ \dot y&=&-x+yz\\ \dot z&=&1-y^2 \end{array}\right.$$ It is chaotic and it has no equilibrium points.