Generalized autoregression and the analysis of dynamical processes

Abstract

The dynamics of nonlinear systems can be characterized in terms of response functions. Mathematical representations of data that contain mixed stochastic-deterministic components are likely to have coefficients that are not well behaved. In these circumstances, statistical averaging can be used to obtain a moment hierarchy that has well behaved coefficients and that can be inverted for the response function values. The moment hierarchy is generated by operating on a convolution series expansion that is truncated so that a tractable set of equations with well behaved coefficients can be solved. The moment heirarchy is used in two contexts. First the analysis of noiseless deterministic mappings is considered as the precursor to the analysis of experimental data. A chaotic numerical example is used to demonstrate the accuracy of the moment heirarchy method to identify the order and from of the mapping and to predict the future behavior of the chaotic sequence. Second, Wolf's annual sunspot number, which is theoretically predicted to be a delayed logistic map, is analyzed and discussed. Finally, experimental data from a driven electronic anharmonic oscillator that exhibits period doubling and chaotic behavior is analyzed and discussed.