Reconstructing bifurcation diagrams with Lyapunov exponents from only time-series data using an extreme learning machine

Abstract

Abstract:We describe a method for reconstructing bifurcation diagrams with Lyapunov exponents for chaotic systems using only time-series data. The reconstruction of bifurcation diagrams is a problem of time-series prediction and predicts oscillatory patterns of time-series data when parameters change. Therefore, we expect the reconstruction of bifurcation diagram could be used for real-world systems that have variable environmental factors, such as temperature, pressure, and concentration. In the conventional method, the accuracy of the reconstruction can be evaluated only qualitatively. In this paper, we estimate Lyapunov exponents for reconstructed bifurcation diagrams so that we can quantitatively evaluate the reconstruction. We also present the results of numerical experiments that confirm that the features of the reconstructed bifurcation diagrams coincide with those of the original ones.

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.…”

“…In this section, we explain how to reconstruct BDs of all components using an ELM only from timeseries data sets. First we explain the ELM as a time-series predictor, and then we explain the algorithm for BD reconstruction using the ELM [8].…”

“…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.…”

A numerical framework for designing periodic orbits embedded in chaotic attractors

“…Owing to the modularity of the proposed framework, it will be interesting to consider various implementation alternatives including e.g. other choices of the underlying structure of chaotic attractors [16], and choices of emerging function approximators such as extreme learning machines that have recently been used for constructing dynamical system models [17,18].…”

A Quantitative Method for Evaluating Reconstructed One-Dimensional Bifurcation Diagrams

“…In addition, they quantitatively evaluate the reconstruction accuracy for two-dimensional BDs. We have previously proposed a method for estimating the Lyapunov spectra of reconstructed BDs [8]. We propose a method for quantitative evaluation of the reconstructed 1-dimensional BDs.…”