Characterization of nonstationary chaotic systems

Nonstationary dynamical systems arise in applications, but little has been done in terms of the characterization of such systems, as most standard notions in nonlinear dynamics such as the Lyapunov exponents and fractal dimensions are developed for stationary dynamical systems. We propose a framework to characterize nonstationary dynamical systems. A natural way is to generate and examine ensemble snapshots using a large number of trajectories, which are capable of revealing the underlying fractal properties of the system. By defining the Lyapunov exponents and the fractal dimension based on a proper probability measure from the ensemble snapshots, we show that the Kaplan-Yorke formula, which is fundamental in nonlinear dynamics, remains valid most of the time even for nonstationary dynamical systems.