Accelerated stability transformation method for chaos control of discrete dynamical systems

Abstract

This paper proposes the accelerated stability transformation method (ASTM) of chaos control to enhance the efficiency of stabilizing the unstable fixed points or periodic orbits of discrete dynamical systems, with fewer iterative number than the original STM. For each step of iteration, the ASTM scheme utilizes the STM formulation twice to control the step size in the oscillation direction and relaxes greatly the step size of direction normal to the oscillation direction and thus reduces computational efforts of chaos control remarkably. Four examples of nonlinear maps including the hyperchaotic system indicate that the proposed ASTM scheme is more efficient and accurate than STM scheme for stabilizing the unstable fixed points embedded in chaotic attractor. The convergence rate of ASTM scheme is closely related to the involutory matrix C and control parameter q. Moreover, two effective selecting rules of involutory matrix C are suggested by analyzing the eigenvalues of Jacobian matrix of the discrete systems.