Machine Learning of Nonlinear Dynamical Systems with Control Parameters Using Feedforward Neural Networks

TL;DR

It is demonstrated that a simpler feedforward neural network can also reproduce the bifurcation diagram of the logistics map and synchronization transition in globally coupled Stuart-Landau equations.

Abstract

Several authors have reported that the echo state network reproduces bifurcation diagrams of some nonlinear differential equations using the data for a few control parameters. We demonstrate that a simpler feedforward neural network can also reproduce the bifurcation diagram of the logistics map and synchronization transition in globally coupled Stuart-Landau equations.

Figures (2)

• Figure 1: (a) The input-output relationship at $a=3.8$. The dashed line represents $y=3.8x(1-x)$. (b) Bifurcation diagram between $2.8\le a\le 4$ by the feedforward network after the learning process of the Ridge regression using the data of the logistic map.
• Figure 2: (a) Order parameter $S$ (solid line) in the coupled equations of the Stuart-Landau oscillators with $M=1000$ as a function of $K$ and the order parameter (dashed line) obtained by the feedforward neural network. (b) Time sequence (solid line) of $X_i(t)$ in the coupled Stuart-Landau equations of $M=3$ and a time sequence of $X_i(t)$ by the three-layer feedforward network with $N=575$ neurons in the intermediate layer (dashed line).