Multiscale measures of phase-space trajectories

<p>Several attempts have been made in characterizing the multiscale nature of fluctuations from nonlinear and nonstationary time series. Particularly, the study of their fractal structure has made use of different approaches like the structure function analysis, the evaluation of the generalized dimensions, and so on. Here we report on a different approach for characterizing phase-space trajectories by using the empirical modes derived via the Empirical Mode Decomposition (EMD) method. We show how the derived Intrinsic Mode Functions (IMFs) can be used as source of local (in terms of scales) information allowing us in deriving multiscale measures when looking at the behavior of the generalized fractal dimensions at different scales. This formalism is applied to three pedagogical examples like the Lorenz system, the Henon map, and the standard map. We also show that this formalism is readily applicable to characterize both the behavior of the Earth’s climate during the past 5 Ma and the dynamical properties of the near-Earth electromagnetic environment as monitored by the SYM-H index.</p>

A Similarity Comparison Method of Homologous Fault Response Fragments under Variable Rotational Speed

Effective fault detection and diagnosis (FDD) for rotating machinery is always a focus issue in improving the prognostic and health management (PHM) of the equipments. The existing usage of similarity measurement has been widely spread in searching the homologous fault responses from vibration signals, but most of them are just suitable for stable speed and cannot be applied in all variable speed conditions. In order to improve measurement performance, a fast-meshed phase portrait (FMPP) frame combining the phase-space technique and box-scoring calculation is proposed. Firstly, the variable-speed signal is divided into multiple undetermined fragments according to fault characteristic orders (FCOs). Secondly, the undetermined fragments are reconstructed into corresponding phase-space trajectories to overcome the time-delay matching inconsistency of the variable speed fragments. Thirdly, the phase-space trajectories are mapped into meshed phase portraits via box-scoring calculation. Such decisive calculation can effectively transforms the diverse unequal fragments into the phase diagrams with same size, which saves time for the subsequent similarity measurement. Finally, the proposed FMPP is tested both for accuracy and timeliness on a self-built bearing bench, where the order tracking (OT) and dynamic time warping (DTW) methods are used for the comparisons. The experiments proved the effectiveness of the proposed method.

Application of Recurrence Quantification Analysis for Non-Linear Dynamical Systems

The Recurrence plots (RPs) have been introduced in several different scientific and medical disciplines. The main purpose of recurrence plot is used to of identify the higher dimensional phase space trajectories. RPs are purely graphically representation which have been designed for the detection of hidden dynamical patterns and non-linearity present in the data, the evaluation of error which is caused by observational noise can be done by Recurrence Quantification Analysis (RQA). RQA method is initially used to minimize the error present in the given signals. RQA method is a basically a technique for the analysis of nonlinear data to quantify the number and duration of a dynamical systems. The recurrence plot is used for time series domain for multidimensional signal also. Recurrence is the property of non-stationary and dynamical system to characteristics the time series analysis in phase space trajectories. Recurrence Quantification Analysis is used to derive from recurrence plots, which are based upon distances matrices of time series.