TITLE: Do chaos and/or limit cycles always require the existence of an unstable fixed point?
QUESTION [7 upvotes]: 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?

REPLY [8 votes]: 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.