A new test for chaos in deterministic systems

Georg A. Gottwald, Ian Melbourne

Abstract

We describe a new test for determining whether a given deterministic dynamical system is chaotic or non–chaotic. In contrast to the usual method of computing the maximal Lyapunov exponent, our method is applied directly to the time–series data and does not require phase–space reconstruction. Moreover, the dimension of the dynamical system and the form of the underlying equations are irrelevant. The input is the time–series data and the output is 0 or 1, depending on whether the dynamics is non–chaotic or chaotic. The test is universally applicable to any deterministic dynamical system, in particular to ordinary and partial differential equations, and to maps. Our diagnostic is the real valued function p(t)=t0(x(s))cos(θ(s))ds, where ϕ is an observable on the underlying dynamicsx(t) and θ(t)=ct+0t(x(s))ds. The constant c > 0 is fixed arbitrarily. We define the mean–square displacement M(t) for p(t) and set K=limt→∞logM(t)/logt. Using recent developments in ergodic theory, we argue that, typically, K=0, signifying non–chaotic dynamics, or K=1, signifying chaotic dynamics.