Bifurcation analysis of a nonlinear pendulum using recurrence and statistical methods: applications to fault diagnostics
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The problem of maintenance of expensive and heavy systems has increased the need for powerful tools to analyze their performance. The methods of recurrence plots (RPs) and statistical measurement have been used as data-driven tools for diagnostics with no possibility of classifying the nature of defect and poor ability to localize it. In order to enhance the efficiency of the forecast, innovative approaches consist of using physics-based features to train a data-based assessment methods. This requires proper analysis of the physical system using appropriate methods. For this purpose, this paper focusses on the bifurcation dynamics of nonlinear systems using the recurrence and statistical methods. Considering the nonlinear pendulum as a model, the qualitative behavior of the system is discussed through the bifurcation diagram of some recurrence quantification analysis (RQA) parameters, namely the recurrence rate, determinism, and laminarity. These parameters are used to measure the level of complexity and transition from regular to chaotic motion and vice versa. Statistical parameters such as crest factor, skewness, and kurtosis are used to identify various bifurcation and amplitudes in the system, and to measure the orientation and the level of asymmetry. Plots of recurrence diagrams and histograms are presented to support our observations. Examples of detection of dynamic changes using these two methods are provided. The interesting results obtained in this paper show that statistical methods complement results obtained from RPs. In addition, the paper demonstrates how the RPs can be employed in conjunction with the physics-based model.
KeywordsTime-series analysis Recurrence analysis Statistical analysis Fault detection
This work is supported by the US Office of Naval Research under the Grant N00014-13-1-0485. We are grateful for this support and deeply appreciate the efforts of Mr. Anthony Seman III of ONR.
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