A novel non-Lyapunov approach through artificial bee colony algorithm for detecting unstable periodic orbits with high orders

In this paper, a novel non-Lyapunov way is proposed to detect the unstable periodic orbits (UPOs) with high orders by a new artificial bee colony algorithm (ABC). And UPOs with high orders of nonlinear systems, are one of the most challenging problems of nonlinear science in both numerical computations and experimental measures. The proposed method maintains an effective searching mechanism with fine equilibrium between exploitation and exploration. To improve the performance for the optimums of the multi-model functions and to avoid the coincidences among the UPOs with different orders, we add the techniques as function stretching, deflecting and repulsion to ABC. The problems of detecting the UPOs are converted into a non-negative functions' minimization through a proper translation, which finds a UPO such that the objective function is minimized. Experiments to different high orders UPOs of 5 wellknown and widely used nonlinear maps indicate that the proposed algorithm is robust, by comparison of results through the ABC and quantum-behaved particle swarm optimization (QPSO), respectively. And it is effective even in cases where the Newton-family algorithms may not be applicable. Density of the orbits are discussed. Simulation results show that ABC is superior to QPSO, and it is a successful method in detecting the UPOs, with the advantages of fast convergence, high precision and robustness.