Multiobjective evolutionary search of difference equations-based models for understanding chaotic systems

In control engineering, it is well known that many physical process exhibit a chaotic component. In point of fact, it is also assumed that conventional modeling procedures disregard it, as stochastic noise, beside nonlinear universal approximators (like neural networks, fuzzy rule-based or genetic programming-based models,) can capture the chaotic nature of the process. In this paper we will show that this is not always true. Despite the nonlinear capabilities of the universal approximators, these methods optimize the one step prediction of the model. This is not the most adequate objective function for a chaotic model, because there may exist many different non-chaotic processes that have near zero prediction error for such an horizon. The learning process will surely converge to one of them. Unless we include in the objective function some terms that depend on the properties on the reconstructed attractor, we may end up with a non chaotic model. Therefore, we propose to follow a multiobjective approach to model chaotic processes, and we also detail how to apply either genetic algorithms or simulated annealing to obtain a difference equations-based model.