Global modeling and prediction of three dimensional systems
C. S. M. Lainscsek and F. Schürrer
Institut für Theoretische Physik,
Technische Universität Graz,
Petersgasse 16,
A-8010 Graz, Austria
Modeling of data in form of differential equations is important in the case when only a reduced set of state variables gathered from measurements of a physical system is accessible.
Characteristic features, e.g., strange attractors, can sometimes be reconstructed from the reduced information on the state of the whole system by using the time delay method. It is expected that a reconstructed attractor is diffeomorph to the attractor in physical phase space. A nonlinear mapping exists between the artificial embedding space and the physical phase space, that can not be extracted from the time series of a reduced set of state variables. If one tries to find a global model from an attractor in the embedding space, the so derived model will in general not describe the behavior in the physical phase space.
A special class of systems of differential equations (e.g., Lorenz Equations and Rössler System) can be transformed to only one three dimensional differential equation, which can be rewritten as system of three ordinary differential equations depending on one variable of the original system. If one tries to model a single variable time series of the original system to such a transformed system, the predicted evolution of the so achieved model is comparable to a model from the whole set of variables of time series.
This method is applied to a simulated time series of the Lorenz- and the Rössler- attractor and the results are compared to the original attractors.