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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?

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  • $\begingroup$ 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). $\endgroup$
    – Steve Heim
    Jan 26, 2018 at 7:54

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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.

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