My first thought was 'let's try it on resting-state fmri data'.
rsfmri data comprise multiple timeseries sampled from voxels throughtout the brain. The dynamics are complex, and there have been attempts to examine it through the lens of attractors etc. I'm not an expert in chaotic analysis, but will say that most of the advances in the field of neuroscience come from innovative analytic methods. Like I said, it was my first thought, but there is a whole subfield examining brain dynamics through these lenses.
That said, another aspect of dissipative chaos is that a highly complex system like turbulence (10^28 or so atoms bouncing off each other) often can be described by an attractor that is rather low dimensional and could be modeled with a few equations. So you might have some neural signal that tools like that could make an interesting model of.