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.

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 a previous example, we demonstrated reconstruction of the BDs of induction motor drives [11]. To apply BD reconstruction to real-world data, we have shown that the BD reconstruction is robust against dynamical and observational noise because time-series data are always affected by noise [12,13]. In the aforementioned studies, the timeseries data sets used for BD reconstruction were generated from electronic circuits.…”

Computing of 3D Bifurcation Diagrams With Nvidia CUDA Technology

“…In order to perform the calculations, special software dedicated to the measuring platform was written, described in detail in the section titled: -Description of the measuring platform.‖ The main solver performing calculations was designed in the form of two separate CUDA kernels (kernel is a function performed on the CUDA device). The task of the first kernel is to solve the system of differential equations using the Runge-Kutta fourth-order method (1).…”

“…This paper uses three-dimensional approach to bifurcation diagrams, while the bifurcation diagrams usually presented in the literature take a single-parameter or much less often twoparameter form [1][2][3][4][5][6]. The use of a three-dimensional approach to bifurcation involves the presentation of a point cloud in space.…”