Unfolding the Phase Space Structure of Noisy Time Series by means of Angular First-Return Maps

A new approach which uses the joint probability matrix computation of noisy time series is proposed to construct a phase space portrait which reflects the orbit visitation frequency of the different regions of the phase space. The resulting representation provides a clear cut of the dynamical reconstructed attractor giving both quantitative information and qualitative information about the attractor structure. The orbital distribution recovered in the map is studied by an angular first-return map where the orbital time used for the reconstruction is obtained from the magnitude information of the complex representation of the data belonging to the probability phase portrait. The resulting phase delay coordinates serve to identify phase intermittency. The Lorenz-like Shimizu-Morioka model and the Rossler model are used to present the methodology. Finally, some experimental pressure time series measured on gas-solid fluidized beds operated at different dynamical regimes are presented to analyze the reliability of the proposed method to deal with experimental noise time series.