Abstract
Bifurcation-diagram reconstruction estimates various attractors of a system without observing all of them but only from observing several attractors with different parameter values. Therefore, the bifurcation-diagram reconstruction can be used to investigate how attractors change with the parameter values, especially for real-world engineering and physical systems for which only a limited number of attractors can be observed. Although bifurcation diagrams of various systems have been reconstructed from time-series data generated in numerical experiments, the systems that have been targeted for reconstructing bifurcation diagrams from time series measured from physical phenomena so far have only been continuous-time dynamical systems. In this paper, we reconstruct bifurcation diagrams only from time-series data generated by electronic circuits in discrete-time dynamical systems with different parameter values. The generated time-series datasets are perturbed by dynamical noise and contaminated by observational noise. To reconstruct the bifurcation diagrams only from the time-series datasets, we use an extreme learning machine as a time-series predictor because it has a good generalization property. Hereby, we expect that the bifurcation-diagram reconstruction with the extreme learning machine is robust against dynamical noise and observational noise. For quantitatively verifying the robustness, the Lyapunov exponents of the reconstructed bifurcation diagrams are compared with those of the bifurcation diagrams generated in numerical experiments and by the electronic circuits.
Highlights
Attractors in real-world systems can be measured as time-series data
We begin by describing the experimental conditions, we show the bifurcation diagram (BD) generated by MATLAB, the electronic circuit, and the reconstructed BD
We begin by describing the experimental conditions, we show the BD generated by MATLAB, the one produced by the electronic circuit, and the reconstructed BD
Summary
Attractors in real-world systems can be measured as time-series data. If the system is exhibiting deterministic chaos, for example, the data can be predicted in the short term by using a predictor that is trained to model the measured time-series data, a target dynamical system itself is usually unknown. If cost and time are limited, a sufficient amount of data often cannot be obtained To address this problem, in 1994, Tokunaga et al. proposed a method of BD reconstruction, whereby the BD of the target system is estimated only from a few time-series datasets measured at different parameter values; by numerical experiments, Tokunaga et al reconstructed the BDs of the Hénon map and coupled logistic/delayed-logistic maps only from time-series data without dynamical and observational noise. Tokuda et al. reconstructed the BD of the Rössler equations only from time-series data with observational noise, and in 2000, Bagarinao et al. reconstructed the BDs of a cubic map and the FitzHugh–Nagumo equations only from time-series data with dynamical noise. In 2001, Small et al. estimated the BD of the Rössler equations only from a time-series dataset with observational noise. Small et al. estimated the BD during the onset of human ventricular fibrillation
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