Question

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...

Answer 1

Here are some easy criteria:

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 , but this is silly, because the motion is to infinity.
As a general rule, they can't occur in 2d phase spaces either, because the orbits form non-intersecting curves which go from repelling critical points to attracting critical points through saddles, dividing the plane into closed regions bound by closed curves which link up nodes and saddles. the curves in the interior of a node-saddle-node-saddle region can be nonintersecting cycles, which means that nearby orbits separate linearly, according to the change of the period along adjacent cycles, or spirals to a stable point, which have zero separation.
In three dimensions, you have chaos. The Lorentz attractor is an example. But these systems are generally dissipative. For a Hamiltonian system, you are restricted to even dimensional phase space, with an energy surface. The double-pendulum provides an example of a chaotic 4 dimensional phase space with a conserved energy, so 3 dimensional constant energy surface.
You don't have chaos in an integrable Hamiltonian system with small enough additional nonlinearities except near resonant points. This is the KAM theorem. The larger the system, the more resonances you have, so the easier it is to be chaotic.
You always have exponential separation of geodesics in negatively curved manifolds. So this is a case where chaos is normal. There are rigorous theorems about the chaos in billiards on negatively curved spaces.