On the Application of DFA to the Analysis of Unimodal Maps

Chaotic time series obtained from simple dynamical systems (the tent map and the logistic map) are analyzed by means of Detrended Fluctuation Analysis (DFA) - a widely used method for quantifying long-range correlations in time series obtained from complex systems. The first conclusion is that time series obtained from stochastic (noise-driven) and deterministic systems may be indistinguishable using the DFA method. We introduce the adaptive DFA exponent and find that it is related to the structure of the periodic orbit. We show that persistence detected in deterministic series by DFA has a different interpretation than that used in the context of stochastic series analysis. For chaotic time series, we find that only a large level of dynamic additive noise can alter the short-range DFA exponent. Finally, a relation between the DFA exponents and the control parameter of the map is studied. The short-range DFA exponent is sensitive to different kinds of nonlinear transitions - we show that the exponent decreases with the merging of chaotic bands and increases as the natural measure becomes more symmetric. If periodic windows occur in the bifurcation diagram, they can be also detected by DFA as an abrupt decrease of the short-range exponent to a value close to 0. An interior crisis occurs at the end of each periodic window - as a result, the DFA exponent increases as a function of the control parameter until the next band-merging point. As the periodic windows are dense in the bifurcation diagram, the relation of the DFA exponent on the control parameter is more complex for this case. (author)