DETECTING LOW-DIMENSIONAL CHAOS IN TIME-SERIES OF FINITE-LENGTH GENERATED FROM DISCRETE PARAMETER PROCESSES

One of the truly novel issues in the physics of the last decade is tha t some time series considered of stochastic origin might in fact be of a particular deterministic type, named ''chaotic''. Chaotic processes are essentially characterized by a low, rather than very high (as in stochastic processes), number of degrees of freedom. There has been a proliferation of attempts to provide efficient analytical tools to dis criminate between chaos and stochasticity, but in most cases their pra ctical utility is limited by the lack of knowledge of their effectiven ess in realistic time series, i.e. of finite length and contaminated b y noise. The present paper attempts to estimate the practical efficien cy of a slightly modified Sugihara and May procedure [G. Sugihara and R.M. May, Nature 344 (1990) 734]. This is applied to synthetic finite time series generated from discrete parameter processes, providing rat es of misidentification (obtained through simulations) for the most co mmon stochastic processes (Gaussian, exponential, autoregressive, and periodic) and chaotic maps (logistic, Henon, biological, Tent, trigono metric, and Ikeda). The procedure consists of comparing with a selecte d threshold the correlation between actual and predicted values one ti me step into the future as a function of the embedding dimension E. Th is procedure allows to infer the presence of low-dimensional chaos eve n on series of similar to 50 units, and in presence of a noise level e qual to similar to 10% of the signal amplitude. We apply this method t o the sequence of volcanic eruptions of Piton de La Fournaise volcano finding no evidence of low-dimensional chaos.