A data-driven method for reconstructing bifurcation diagrams

Abstract

In this study, we proposed a method for reconstructing one-parameter bifurcation diagram of a dynamical system using time-series data. Logistic map and FitzHugh-Nagmo (FHN) model were used to validate our method. For each system, we generated synthetic time-series data with various bifurcation parameter values, and then estimated vector fields (or map) of the systems using multilayer neural networks. Next, we estimated bifurcation parameters based on a principal curve approach from the weight-space of the learned networks, and finally reconstructed their bifurcation diagrams. Using the proposed algorithm, we could reconstruct satisfactory bifurcation diagrams which preserve the important bifurcation structures of the original systems, such as the period-doubling bifurcation or chaos for the logistic map, and the hopf bifurcation or limit cycles for the FHN model. We further discuss the effects of data length, observation noises, or number of nodes in hidden layers.