Machine learning, alignment of covariant Lyapunov vectors, and predictability in Rikitake's geomagnetic dynamo model

In this paper, the alignment of covariant Lyapunov vectors is used to train multi-layer perceptron ensembles in order to predict the duration of regimes in chaotic time series of Rikitake's geomagnetic dynamo model. The machine learning procedure reveals the relevance of the alignment of distinct covariant Lyapunov vectors for the predictions. To train multi-layer perceptron, we use a classification procedure that associates the number of maxima (or minima) inside regimes of motion with the duration of the corresponding regime. Remarkably accurate predictions are obtained, even for the longest regimes whose duration times are around 17.5 Lyapunov times. We also found long duration regimes with a distinctive statistical behavior, namely, the longest regimes are more likely to occur, a quite unusual behavior. In fact, we observed a largest regime above which no regimes were observed.