I found this paper of Cuomo and Oppenheim, where they use a Lorenz system to define an encryption scheme for signals. There is also this blog post describing and implementing the technique. The technique uses an interesting property of the Lorenz system known as "synchronization," which as far as I know occurs in other dynamical systems as well. Moreover, the chaotic nature of these systems intuitively makes guessing the parameters of the system infeasible, so a natural shared secret key is the set of constants used to define the system.
I was wondering if there has been any theoretical work in cryptography that deals with dynamical systems. Specifically, are there established cryptographic hardness assumptions for specific dynamical systems? It seems like it would be a different flavor of hardness assumption from the usual ones, because the methods naturally introduce some noise into the decrypted message.