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?
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.