Stochastic versions of chaotic time series: Generalized logistic and Hénon time series models

Abstract

Deterministic time series models, particularly those which exhibit chaotic behavior, have received considerable attention recently as an alternative to stochastic time series models. In many cases there exists a well-defined stochastic time series model from which the deterministic time series model is obtained in the limit as the degree of stochasticity goes to zero. The stochastic version may then be used as a means for fitting the deterministic model. In this paper we introduce a stochastic generalized (parametric) logistic time series model. Using simulated data, we show that model fitting methods developed for the stochastic logistic time series also work in the deterministic case. We then use these methods to fit the logistic model to the weevil data of Utida. We introduce as well a stochastic quadratic time series model for which the deterministic Hénon time series model is a limiting special case. Finally, we show how the stochastic version of a nonlinear deterministic time series model provides a mechanism for studying the stability of the deterministic model to small perturbations.