Proc. Roy. Soc. London A 460 (2004) 603-611.
Georg Gottwald and Ian Melbourne
Abstract
We describe a new test for determining whether a given deterministic dynamical system is chaotic or nonchaotic. 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 is 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) = \int_0^t F(x(s))cos(\theta(s))ds
where F is an observable on the underlying dynamics x(t) and
\theta(t) = ct + \int_0^t F(x(s))ds.
The constant c>0 is fixed arbitrarily. We define the mean-square-displacement M(t) for p(t) and set
K = \lim_{t \rightarrow \infty} log M(t)/log t.
Using recent developments in ergodic theory, we argue that typically K=0
signifying nonchaotic dynamics or K=1 signifying chaotic dynamics.