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

Abstract

It has been quite common in the analysis of single-or multi-fractal signals originating from complex non-linear systems to make a time-delayed construction of the state space attractor in which the dynamics can be qualitatively viewed. This involves the calculations of the embedding dimension and an appropriate time delay based on the signal non-linear correlation behavioural pattem. This is usually followed by a sub-optimal short-term linear prediction in the signal timeisubspace domain instead of the optimal non-linear prediction in the timeifrequency domain. In a previous paper, we proposed a new family of embedded multiple-step Volterra structures that exploit past and present non-linear signal dynamics seeded in the attractor and integrate them in their design. Essentially, this is done by including relevant past information preserved in the signal up to 2 time td = the embedded dimension X embedded time delay, and sampled at instances coincident with the embedded time delays, to predict one-step ahead, adaptively, using the LMS criteria. This resulted in superior recursive prediction performance (up to 30 dBs improvement in the MSE) over the conventional Volterra structures, when individually optimiscd in their characteristic parameters and tested on non-linear and chaotic mono-fractal real systems. The adaptive algorithm is highly amenable tu parallel implementation as it is designed tu cater fur new parallel Volterra structures, with progressively optimised delayed inputs. However, when tested on highly dynamic multi-fractal chaotic systems the embedded Volterra structure was marginally better than the conventional one, both could only perform short-term non-recursive prediction. In this paper, we specifically tum our attention to long-term prediction of this particular class of multi-fractal chaotic systems. Here, the linear, quadratic, cubic, and nthurder 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 close6loop adaptation sdkeme without any external input signal applied tu the structure. The dynamic invariants computed from the reconstructed t i e series must now closely match the corresponding ones coniputed 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 here, interesting results obtained using Loren2 attractor, and performing two r...