Toward a Test for Departure of a Trajectory from a Neighborhood of a Chaotic System
Many real systems have been identified as potentially chaotic (e.g., trains of heart beats, records of multiple populations in an ecological system, weather and climate data). In such cases, when “healthy” is defined by the measured trajectory being confined to a “strange attractor” in a “phase space,” how can we define and measure degradation? The simplest course is to define degradation as when the attractor changes. The earliest warning of such a change would be when a trajectory leaves an attractor, without an external forcing, but there seems little written on the problem of identifying whether a short trajectory belongs to a given attractor (at least to within a neighborhood defined by stochastic disturbances and uncertainty in the parameters of the defining equations).
We consider a system x(t) = (x 1(t),…, x N (t)) consisting of N Brownian particles with synchronizing interaction between them occurring at random time moments . Under assumption that the free Brownian motions and the sequence are independent we study asymptotic properties of the system when both the dimension N and the time t go to infinity. We find three time scales t = t(N) of qualitatively different behavior of the system.