In this paper we report on generalized Lorenz models. Five-and six-dimensional Lorenz models are investigated, which are obtained by considering respectively two and three additional Fourier modes in addition to the modes included in the derivation of the classical three-dimensional Lorenz model. Parameter planes, bifurcation diagrams, and attractors in the phase-space are used, in order to investigate the influence of the additional Fourier modes on solutions, when compared with the solutions for the classical Lorenz model. It is shown that for parameters σ and b kept fixed, a larger parameter r results for the onset of chaos in five-and six-dimensional Lorenz models. Also it is shown that the shape of bifurcation diagrams, periodic, and chaotic attractors is preserved in both generalized Lorenz models. Additionally, it is shown that hyperchaos is observed only in the six-dimensional Lorenz model, at least in the parameter ranges here investigated.
This work considers the problem of predicting wing changes, and their duration, in systems able to support three interconnected wings (spirals) in the space of the variables. This is done by exploring the alignment of covariant Lyapunov vectors (CLVs) known to precede the occurrence of peaks and regime changes in some chaotic systems. Here, the alignment of the CLVs is combined with classification procedures, machine learning techniques and nearest-neighbors analysis to predict the duration inside coupled wings. The classification procedure has distinct classes defined as the number of maxima of one variable inside one wing, proportional to the time spent inside each wing. In general, we observe that predictions of significant duration times (around [Formula: see text] oscillations) inside a wing can be efficiently predicted until four subsequently visited wings. For five to ten subsequently visited wings, the prediction efficiency decreases significantly for more than ten oscillations inside each wing. The numerical comparison shows the superiority of the nearest-neighbors methods over the multilayer perceptrons procedure for the predictions. Remarkably, accuracies larger than [Formula: see text] are obtained for predictions of classes for [Formula: see text] subsequently visited wings. Accuracies higher than [Formula: see text] are obtained for predictions of classes in the [Formula: see text] subsequently visited wings.
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