Itoh Y, Uenohara S, Adachi M, Morie T, Aihara K
Department of Electrical and Electronic Engineering, Hokkaido University of Science, Hokkaido 006-8585, Japan.
Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan.
Chaos. 2020 Jan;30(1):013128. doi: 10.1063/1.5119187.
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.
分岔图重建可在不观测系统所有吸引子的情况下，仅通过观测具有不同参数值的几个吸引子来估计系统的各种吸引子。因此，分岔图重建可用于研究吸引子如何随参数值变化，特别是对于那些只能观测到有限数量吸引子的实际工程和物理系统。尽管已从数值实验中生成的时间序列数据重建了各种系统的分岔图，但迄今为止，从物理现象测量的时间序列重建分岔图所针对的系统仅为连续时间动态系统。在本文中，我们仅从具有不同参数值的离散时间动态系统中的电子电路生成的时间序列数据重建分岔图。生成的时间序列数据集受到动态噪声的干扰并被观测噪声污染。为了仅从时间序列数据集重建分岔图，我们使用极限学习机作为时间序列预测器，因为它具有良好的泛化特性。由此，我们期望使用极限学习机进行的分岔图重建对动态噪声和观测噪声具有鲁棒性。为了定量验证鲁棒性，将重建分岔图的李雅普诺夫指数与数值实验和电子电路生成的分岔图的李雅普诺夫指数进行比较。
