Comparison of systems with complex behavior

We present a formalism for comparing the asymptotic dynamics of dynamical systems with physical systems that they model based on the spectral properties of the Koopman operator. We first compare invariant measures and discuss this in terms of a “statistical Takens” theorem proved here. 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 in this paper 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 a class of models that have the form of a random dynamical system.