Nonlinear forecasting of non-uniform chaotic attractors in an enzyme reaction

Abstract

Nonlinear forecasting was used to predict the time evolution of fluctuating concentrations of dissolved oxygen in the peroxidase-oxidase reaction. This reaction entails the oxidation of NADH with molecular oxygen as the electron acceptor. Depending upon the experimental conditions, either regular or highly irregular oscillations obtain. Previous work suggests that the latter fluctuations are almost certainly chaotic. In either case, the dynamics contain multiple timescales, which fact results in an uneven distribution of points in the phase space. Such ‘nonuniformity,’ as it is called, is a rock on which conventional methods for analysing chaotic time series often founder. The results of the present study are as follows. 1. Short-term forecasting with local linear predictors yields results that are consistent with a hypothesis of low-dimensional chaos. 2. Most of the evidence for nonlinear determinism disappears upon the addition of small amounts of observational error. 3. It is essentially impossible to make predictions over time intervals longer than the average period of oscillation for time series subject to continuous and frequent sampling. 4. Far more effective forecasting is possible for points on Poincare sections. 5. An alternative means for improving forecasting efficacy using the continuous data is to include a second variable (NADH concentration) in the analysis. Since non-uniformity is common in biological time series, we conclude that the application of nonlinear forecasting to univariate time series requires care both in implementation and interpretation.