Period Doubling Cascades from Data

Orbit diagrams of period doubling cascades represent systems going from periodicity to chaos. Here, we investigate whether a Gaussian process regression can be used to approximate a system from data and recover asymptotic dynamics in the orbit diagrams for period doubling cascades. To compare the orbits of a system to the approximation, we compute the Wasserstein metric between the point clouds of their obits for varying bifurcation parameter values. Visually comparing the period doubling cascades, we note that the exact bifurcation values may shift, which is confirmed in the plots of the Wasserstein distance. This has implications for studying dynamics from time series data. Although the accuracy is good away from bifurcation points, an approximation of a system’s period doubling cascade may lead to unpredictable model behavior in a neighborhood of the true bifurcation parameter value