INDUCED ONE-PARAMETER BIFURCATIONS IN IDENTIFIED NONLINEAR DYNAMICAL MODELS

Abstract

It is shown that nonlinear global models identified from a single time series can be used to reproduce the same sequence of bifurcations of the original system. This has been observed for simulated and real data and for both difference equation and differential equation models, thus suggesting some generality. The results reported in this paper are of a practical character and seem to have some bearing not only on the important subject of estimating bifurcation diagrams from data, but also in model validation problems since some models can reproduce the bifurcation sequence of a system even when such models do not settle to the original attractor at first. In this case, models which would be dismissed are shown to display consistent dynamic information about the original system, as illustrated by a simulated and a real data example. An additional example that uses real data is provided in which the original bifurcation sequence is recovered by the addition of multiplicative noise with increasing variance.