Detecting Unstable Periodic Orbits in Hyperchaotic Systems Using Subspace Fixed-Point Iteration

We present a numerical method for efficiently detecting unstable periodic orbits(UPO’s)embedded in chaotic attractors of high-dimensional systems.This method,which we refer to as subspace fixed-point iteration, locates fixed points of Poincare maps using a form of fixed-point iteration that splits the phase space into appropriate subspaces.In this paper,among a number of possible implementations,we primarily focus on a subspace method based on the Schmelcher-Diakonos(SD)method that selectively locates UPO’s by varying a stabilizing matrix,and present some applications of the resulting subspace SD method to hyperchaotic attractors where the UPO’s have more than one unstable direction.

Harmonic balance-based approach for optimal time delay to control unstable periodic orbits of chaotic systems

As a classical technique for chaos suppression,the time-delayed feedback controlling strategy has been widely developed by stabilizing unstable periodic orbits(UPOs)embedded in chaotic systems.A critical issue for achieving high controlling precision is to search for an appropriate time delay.This paper proposes a simple yet effective approach,based on incremental harmonic balance method,to determine the optimal time delay in the delayed feedback controller.The time delay is adjusted within the iterative scheme provided by the proposed method,and finally converges to the period of the target UPO.As long as the optimal time delay is fixed,moreover,the attained solution makes it quite convenient to analyze its stability according to the Floquet theory,which further provides the effective interval of the feedback gain.