A Quantitative Method for Evaluating Reconstructed One-Dimensional Bifurcation Diagrams

Abstract

We describe a quantitative method for evaluating reconstructed 1-dimensional bifurcation diagrams. We estimate the oscillatory patterns of time-series data by reconstructing the bifurcation diagrams from time-series data alone. Such reconstruction can be used for real-world systems that have variable parameters, such as electric current and power, temperature, pressure, and concentration. In the conventional method, the reconstructed bifurcation diagram is qualitatively compared with the original one. Here, we evaluate the reconstructed 1-dimensional bifurcation diagrams by means of quantitative comparison with the original one. We also present the results of numerical experiments, demonstrating that our method is useful for quantitative evaluation of reconstructed bifurcation diagrams for the Hénon map and the Rössler equations.

Reconstructing one- and two-bifurcation diagrams of all components in the Rössler equations only from time-series data sets

“…In the BD reconstruction, time-series predictors are trained to model the obtained time-series data sets of all components, and the method allows the attractors of all components to be estimated when the bifurcation parameter values are changed. We reconstructed the BDs for various systems and numerical conditions using time-series data sets of a component in [8][9][10][11][12][13][14]. In this study, we show that the reconstruction requires a shorter length of training data when using time-series data sets of all components than when using the time series of only one component, compared by the previous studies.…”

“…Given its extremely high training speed and good generalization, we use the ELM as a time-series predictor. In previous work, we showed that the ELM is useful for reconstructing BDs [8][9][10][11][12][13].…”

“…In 1994, Tokunaga et al proposed a method for reconstructing bifurcation diagrams (BDs) [1] that several research groups then studied [2][3][4][5][6][7][8][9][10]. BD reconstruction allows attractors to be estimated when the bifurcation parameter values of a target system are changed.…”