A new class of non-linear, multi-dimensional structures for long-term dynamic modelling of chaotic systems

Abstract

In this paper, we specifically turn our attention to long-term prediction of dynamic multi-fractal chaotic systems. Here, the linear, quadratic, cubic, and nth-order non-linearities are each multiplied by a weighting function. The weighting functions can take a time-varying form, if necessary, to cater for the non-stationary dynamics of the signal. During the training phase, the characteristic parameters of the weighting functions adapt to the varying nature and emphasis of non-linearity. Once the training of the new adaptive structure is completed; the generalization performance is evaluated by performing recursive prediction in an autonomous fashion. Specifically, the long-term predictive capability of the structure is tested by using a closed-loop adaptation scheme without any external input signal applied to the structure. The dynamic invariants computed from the reconstructed time series must now closely match the corresponding ones computed from the original time series. We will provide evidence of long-term prediction in excess of several thousand samples of highly complex (nine dimension) multi-fractal labour contraction signals using only a small fraction of this sample (only 300 samples for the training phase). Also presented are interesting results obtained using Lorenz attractor, and performing two recursive long-term predictions; (i) the regularized Gaussian radial basis function networks, and (ii) our novel embedded Volterra-like structure with weighted linear, quadratic and cubic nonlinearities, which demonstrate the superior performance of the latter with reduced SNRs.