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The study of non-linear oscillator chains in classical many-body dynamics hasa storied history going back to the seminal work of Fermi, Pasta, Ulam andTsingou (FPUT). We introduce a new family of such systems which consist ofchains of $N$ harmonically coupled particles with the non-linearity introducedby confining the motion of each individual particle to a box/stadium with hardwalls. The stadia are arranged on a one dimensional lattice but theyindividually do not have to be one dimensional thus permitting the introductionof chaos already at the lattice scale. For the most part we study the casewhere the motion is entirely one dimensional. We find that the system exhibitsa mixed phase space for any finite value of $N$. Computations of Lyapunovspectra at randomly picked phase space locations and a direct comparisonbetween Hamiltonian evolution and phase space averages indicate that theregular regions of phase space are not significant at large system sizes. Whilethe continuum limit of our model is itself a singular limit of the integrablesinh-Gordon theory, we do not see any evidence for the kind of non-ergodicityfamously seen in the FPUT work. Finally, we examine the chain with particlesconfined to two dimensional stadia where the individual stadium is alreadychaotic, and find a much more chaotic phase space at small system sizes.

Ergodic and non-ergodic many-body dynamics in strongly nonlinear lattices