Simultaneous modeling of nonlinear deterministic and stochastic dynamics

We present a method which can simultaneously model the nonlinear deterministic and stochastic dynamics underlying an observed time series. It is formulated to treat Markov processes in continuous and discrete time. The procedure, which we call Sequin (Stochastic EQUation INference) is a generalization of that which we have developed previously for continuous time systems (Borland and Haken, 1992a,b; 1993a,b; Haken 1988). The theory behind the method as well as its numerical implementation is discussed. Simulations using Sequin to analyze finite-size time series stemming from nonlinear processes with additive and multiplicative dynamical noise in continuous time are shown. Furthermore, we show that Sequin can extract the underlying dynamics of discrete time noisy chaotic processes, such as the logistic map with additive dynamical noise. The success of the method in being able to charaterize a system consisting partly of a deterministic chaotic element, and partly of a stochastic element indicates its possible application to many interesting real-world problems.