Stabilization of Higher Periodic Orbits of the Duffing Map using Meta-evolutionary Approaches: A Preliminary Study

This paper deals with an advanced adjustment of stabilization sequences for complex chaotic systems by means of meta-evolutionary approaches in the form of a preliminary study. In this study, a two-dimensional discrete-time dynamic system denoted as Duffing map, also called Holmes map, was used. In general, the Duffing oscillator model represents a real system in the field of nonlinear dynamics. For example, an excited model of a string choosing between two magnets. There are many articles on the stabilization of various chaotic maps, but attempts to stabilize the Duffing map, moreover, for higher orbits, are rather the exception. In the case of period four, this is a novelty. This paper presents several approaches to obtaining stabilizing perturbation sequences. The problem of stabilizing the Duffing map turns out to be difficult and is a good challenge for metaheuristic algorithms, and also as benchmark function. The first approach is the optimal parameterization of the ETDAS model using multi-restart Nelder-Mead (NM) algorithm na Genetic Algorithm (GA). The second approach is to use the symbolic regression procedure. A perturbation model is obtained using Genetic Programming (GP). The third approach is two-level optimization, where the best GP model is subsequently optimized using NM and GA algorithms. A novelty of the approach is also the effective use of the objective function, precisely in relation to the process of optimization of higher periodic paths.