Retrieving Dynamical Invariants from Chaotic Data Using NARMAX Models

This paper is concerned with the estimation of dynamical invariants from relatively short and possibly noisy sets of chaotic data. In order to overcome the difficulties associated with the size and quality of the data records, a two-step procedure is investigated. Firstly, NARMAX models are fitted to the data. Secondly, such models are used to generate longer and cleaner time sequences from which dynamical invariants such as Lyapunov exponents, correlation dimension, the geometry of the attractors, Poincare maps and bifurcation diagrams can be estimated with relative ease. An additional advantage of this procedure is that because the models are global and have a simple structure, such models are amenable for analysis. It is shown that the location and stability of the fixed points of the original systems can be analytically recovered from the identified models. A number of examples are included which use the logistic and Henon maps, Duffing and modified van der Pol oscillators, the Mackey-Glass delay system, Chua's circuit, the Lorenz and Rossler attractors. The identified models of these systems are provided including discrete multivariable models for Chua's double scroll, Lorenz and Rossler attractors which are used to reconstruct the trajectories in a three-dimensional state space.