Detecting low-dimensional chaos in time series of finite length generated from discrete parameter processes

Abstract

One of the truly novel issues in the physics of the last decade is that 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 discriminate between chaos and stochasticity, but in most cases their practical utility is limited by the lack of knowledge of their effectiveness in realistic time series, i.e. of finite length and contaminated by noise. The present paper attempts to estimate the practical efficiency 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 rates of misidentification (obtained through simulations) for the most common stochastic processes (Gaussian, exponential, autoregressive, and periodic) and chaotic maps (logistic, Hénon, biological, Tent, trigonometric, and Ikeda). The procedure consists of comparing with a selected threshold the correlation between actual and predicted values one time step into the future as a function of the embedding dimension E. This procedure allows to infer the presence of low-dimensional chaos even on series of ∼ 50 units, and in presence of a noise level equal to ∼ 10% of the signal amplitude. We apply this method to the sequence of volcanic eruptions of Piton de La Fournaise volcano finding no evidence of low-dimensional chaos.