Stabilization of Higher Periodic Orbits of Chaotic maps using Permutation-selective Objective Function

Abstract

This paper deals with the design of an advanced objective function capable of proper evaluation of the solutions during the process of chaotic trajectory stabilisation into stable periodic motion by means of evolutionary metaheuristic optimization. The challenging problem of stabilisation of chaotic systems generates many unexpected difficulties. One of them is the evaluation of a sample stabilized run during optimization. Even more so, when the target state of the chaotic system is a stable cycle oscillating periodically between several target positions. In this study, a two-dimensional dynamical system, known as the Hénon map was used. The system is stabilized using Extended Time Delayed Auto Synchronization (ETDAS) method with Genetic Algorithm (GA) optimization. The solutions are evaluated by a permutation-selective objective function, which achieves significantly better results than conventional evaluation methods based on a common objective function.