Reconstructing bifurcation diagrams only from time-series data generated by electronic circuits in discrete-time dynamical systems

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.