Abstract
For the multi-dimensional chaotic mapping system with various forms of bifurcations, it is found that the spectral radius around the static bifurcation point is approximately equal to one and the convergence speed for traditional stability transformation method (STM) which is used to stabilize the unstable fixed points is fairly slow. In this paper, a modified STM is proposed to efficiently stabilize a 3D chaotic mapping system to stable fixed points. Firstly, according to the information of the fixed point, the stability matrix C is derived, demonstrating that it is unnecessarily an involutory matrix for STM that can also stabilize chaotic system to the fixed point. Then, the critical parameter qcrit that satisfies convergence condition and the optimal parameter qopt corresponding to best performance for STM are determined respectively. Moreover, STM is combined with Newton method (NT) to overcome the disadvantage of slow convergence around the static bifurcation point without requiring a priori information of fixed point. It is indicated that the number of iterations, the absolute and relative errors between the convergent value and analytical fixed point for combined STM-NT decrease enormously comparing with that of traditional STM. Finally, numerical analysis verifies the high efficiency of modified STM proposed in this paper.
Highlights
Chaos is an intrinsically complex phenomenon for deterministic nonlinear dynamical system
stability transformation method (STM) has been widely used in chaos control, structure reliability and optimization design, the convergence speed is very slow when stabilized fixed points are around the static bifurcation points of chaotic system
In this paper a novel modified STM is proposed to efficiently stabilize a 3D chaotic mapping system with various forms of bifurcations to stable fixed points. Focusing on these problems aforementioned, this paper provides some new insights on stabilizing and controlling the unstable periodic fixed points of a 3D chaotic mapping system with various forms of bifurcations by modified STM
Summary
Chaos is an intrinsically complex phenomenon for deterministic nonlinear dynamical system. From another perspective, Pyragas, et al proposed and subsequently developed time-delayed feedback control (DFC) in a series of papers [5,6,7] to detect UPOs by stabilizing continuous chaotic system to period orbit. Parsopoulos et al [8, 9] applied the stochastic optimization method (say, particles warm optimization) to locate chaotic system to high periodic orbits Because this algorithm does not need the detailed information of initial condition and partial derivatives, it is an efficient alternative. STM has been widely used in chaos control, structure reliability and optimization design, the convergence speed is very slow when stabilized fixed points are around the static bifurcation points of chaotic system. The aim of this work is to deepen the understanding for STM and promote its practical application
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