# Chaos and continuous flow

What needs to be the case for a dynamical system with a continuous flow to exhibit chaos? It looks like 1D systems with a continuous flow can't exhibit chaos. Are two dimensions enough or do you need more? I was just thinking about what sort of phase portraits you could have in two dimensions, and it's not immediately obvious if they will always have stable attractors...

• So is the real question here about the minimum dimensionality of the phase space, rather than other properties (like Lyapunov exponents) that are part of the definition of Chaos? – Roy Simpson Feb 28 '11 at 16:59
• Well, dimensionality is my main interest, but I'm interested in what else has to be the case. Put it this way: things like positive (or is it negative? I forget) lyapunov exponents, "sensitive dependence" and the like are what I'd take as characterising a system as chaotic. So the question should be "What needs to be the case for it to be possible that a dynamical system can have positive lyapunov exponents?" – Seamus Mar 1 '11 at 13:14
• I think this is not physics! – Georg Sep 18 '11 at 11:35

• As a general rule, positive Lyapunov exponents can't occur in bounded 1d systems, because the orbits are along a line, and go from repelling fixed points to attracting fixed points, so they cannot separate. The counterexample is $\dot{y}=y$, but this is silly, because the motion is to infinity.