Deep learning in bifurcations of particle trajectories

Abstract

We show that deep learning algorithms can be deployed to study bifurcations of particle trajectories. We demonstrate this for two physical systems, the unperturbed Duffing equation and charged particles in magnetic reversal by using the AI Poincar\'e algorithm. We solve the equations of motion by using a fourth-order Runge-Kutta method to generate a dataset for each system. We use a deep neural network to train the data. A noise characterized by a noise scale L is added to data during the training. By using a principal component analysis, we compute the explained variance ratios for these systems which depend on the noise scale. By plotting explained ratios against the noise scale, we show that they change at bifurcations. For different values of the Duffing equation parameters, these changes are of the form of different patterns of growth-decline of the explained ratios. For the magnetic reversal, the changes are of the form of a change in the number of principal components. We comment on the use of this technique for other dynamical systems with bifurcations.