Tracking bifurcation curves in the Hénon map from only time-series datasets

Abstract

We describe a method for tracking bifurcation curves from only time-series datasets. We apply a tracking algorithm to unknown systems based on the reconstruction of bifurcation diagrams that can estimate the parameter space and oscillatory patterns when parameters change. Therefore, this method can track the bifurcation curves of unknown systems from only measured time-series datasets, whereas the target systems in previous studies are known. By tracking bifurcation curves, we can obtain bifurcation points with increased accuracy as compared with bifurcation diagrams plotted by brute-force methods. In this paper, we present the results of numerical experiments in which bifurcation curves of a Hénon map as an unknown system are tracked from only several time-series datasets.