Comparison of systems with complex behavior: spectral methods

Abstract

We present a formalism for comparing the asymptotic dynamics of dynamical systems with the physical systems that they model. There is often no need for the detailed (trajectory-wise) comparison of a dynamical system and the physical system that it models, but only comparison in statistical sense. For that purpose, invariant measures are typically considered. But, invariant measures usually can not be observed directly in an experiment. Thus, we base our formalism on time-averages obtained from a single observable. In particular, we constructively prove that, generically, a single observable is needed in order to recover an invariant ergodic measure. Pseudometrics on the space of dynamical systems can be defined using this formalism in order to compare their statistical behavior. We also identify the need to go beyond comparing only invariant ergodic measures of systems and introduce an ergodic-theoretic treatment of a class of spectral functionals that allow for this. The formalism is extended for a class of stochastic systems: discrete random dynamical systems. The ideas introduced can be used for parameter identification and model validation of driven nonlinear models with complicated behavior. As an illustration we provide an example in which we compare the asymptotic behavior of a combustion system measured experimentally with the asymptotic behavior of the model that is a stochastic control dynamical system.