Modeling and synchronizing chaotic systems from time-series data

The problem of obtaining ordinary differential equations (ODE's) from chaotic time-series data is addressed. The vector fields for the ODE's are polynomials constructed from a basis set that is orthonormal on the data. The method for constructing the model is similar to the integration of ODE's using Adams predictor-corrector integration. The method is compared to the usual Euler model and is shown to be accurate for much larger sampling intervals. In addition, the method used to construct the model is capable of determining the optimal polynomial vector field for the given data. Finally, we demonstrate that it is possible to synchronize (in the sense of Fujisaka and Yamada Prog. Theor. Phys. 69, 32 (1983) as well as Pecora and Carroll Phys. Rev. Lett. 64, 821 (1990); Phys. Rev. A 44, 2374 (1991)) the model to a time series. Synchronization is used as a nontrivial test to determine how close the model vector field is to the true vector field. Implications and possible applications of synchronization are discussed.