For highD, best way is to "visualize" such oscillators as traveling/moving on a n-torus in the phase space. (they can be quasi-periodic etc., which is its own thing including as a transition to chaos and intermittency).
With regards to the Rajan et al., paper, we can suppress chaos in high dimensional RNNs quite well even without invoking stimulus dependency (i.e., driven activity), a global "adaptation level" with competitive dynamics can avert chaos in autonomous systems.
Here's the introduction from a great paper from my PhD advisor from 1978, which I believe should be cited more:
"The following problem, in one form or another, has intrigued philosophers and scientists for hundreds of years: How do arbitrarily many individuals,
populations, or states, each obeying unique and personal laws, ever succeed in harmoniously interacting with each other to form some sort of stable society, or collective mode of behavior? Otherwise expressed, if each individual obeys complex laws, and is ignorant of other individuals except via locally received
signals, how is social chaos averted? How can local ignorance and global order,
or consensus, be reconciled? This paper considers a class of systems in which this dilemma is overcome."
NicoleCRust