The configuration space for all possible positions of a double pendulum can be represented by $S^1\times S^1 = \mathbb{T}^2,$ i.e. it is represented by a 2 dimensional surface. This leads me to think that (loosely) "chaotic" systems are just systems for which the configuration space is at least 2 dimensional.
>But there are plenty of simple systems where the configuration space is of a higher dimension
Yes, however once we take constraints into account, we can lower the dimension of the actual configuration space to one dimension.

Am I on the right track here? Are there counterexamples, and if so, is there a way to refine this idea?