Necessity of statistical modeling of deterministic chaotic systems

Abstract

The success of deterministic modeling of a physical system relies on whether the solution of the model would approximate the dynamics of the actual system. When the system is chaotic, situations can arise where periodic orbits embedded in the chaotic set have distinct number of unstable directions, a dynamical property known as unstable dimension variability. As a consequence, no model of the system produces reasonably long trajectories that are realized by nature. We argue and present physical examples indicating that, in such a case, though the model is deterministic and low-dimensional, statistical quantities can still be reliably computed. We also argue that unstable dimension variability may be common in high-dimensional chaotic systems such as those arising from discretization of nonlinear partial differential equations.