Computing Chaotic Time-Averages from a Small Number of Periodic Orbits

Temporal averages are useful quantities in ergodic systems, since they are invariant to how the system is initialized and depend only on the structure of the chaotic set. As a result, temporal averages are used frequently in physics and engineering, for instance, in design optimization problems. In chaotic systems, temporal averages can often be approximated using collections of periodic orbits--unstable time-periodic solutions of the governing equations--embedded within the chaotic set. This connection is formalized by periodic orbit theory, with a large number of such solutions required to obtain a good approximation. Here, we describe an alternative, data-driven approach that allows for an accurate approximation of temporal averages even when the number of time-periodic solutions used is quite small. Moreover, this approach yields convergence to temporal averages, in terms of the number of solutions used, that far outperforms periodic orbit theory.