Deterministic, chaotic dynamical systems, when followed through time, have the potential to produce sequences of data that not only appear random but that also rigorously qualify as random from some points of view. A few years ago, Mark Berliner initiated a thorough study of these systems, and I've joined in on some of the work.
This paper takes a look at the notion that chaos is equivalent to paths of a dynamical system separating at an exponential rate. It shows how many paths of the system may actually converge to each other while the set of paths that do converge to each other have Lebesgue measure 0.
Dynamical systems typically produce massive amounts of data. That means that a primary concern is what we can say about such a system based on large (or even asymptotically large) samples. The first paper below describes what can and what can't be determined asymptotically when a system is observed with measurement error. The second paper focuses on the connection between information and consistency, illustrated with a dynamical system.
We often prefer to model a system as having a modest stochastic component as well as a driving deterministic component. Even a small stochastic component can have a big impact on the qualitative behavior of a dynamical system. The paper below illustrates some of the possibilities and makes use of an interesting technique from computer science called interval analysis.