一种具有隐藏吸引子的分数阶混沌系统的动力学分析及有限时间同步

Dynamic analysis and finite time synchronization of a fractional-order chaotic system with hidden attractors

对于具有隐藏吸引子的混沌系统,既有文献大多只针对整数阶系统进行分析与控制研究.基于Sprott E系统,构建了仅有一个稳定平衡点的分数阶混沌系统,通过相位图、Poincare映射和功率谱等,分析了该系统的基本动力学特征.结果显示,该系统展现出了丰富而复杂的动力学特性,且通过随阶次变化的分岔图可知,系统在不同阶次下呈现出周期运动、倍周期运动和混沌运动等状态,这些动力学特征对于保密通信等实际工程领域有重要的研究价值.针对该具有隐藏吸引子的分数阶系统,应用分数阶系统有限时间稳定性理论设计控制器,对系统进行有限时间同步控制,并通过数值仿真验证了其有效性.

Shilnikov criteria believe that the emergence of chaos requires at least one unstable equilibrium, and an attractor is associated with the unstable equilibrium. However, some special chaotic systems have been proposed recently, each of which has one stable equilibrium, or no equilibrium at all, or has a linear equilibrium（infinite equilibrium）. These special dynamical systems can present chaotic characteristics, and the attractors in these chaotic systems are called hidden attractors due to the fact that the attraction basins of chaotic systems do not intersect with small neighborhoods of any equilibrium points. Since they were first found and reported in 2011, the dynamical systems with hidden attractors have attracted much attention. Additionally, the fractional-order system, which can give a clearer physical meaning and a more accurate description of the physical phenomenon, has been broadly investigated in recent years. Motivated by these two considerations, in this paper, we propose a fractional-order chaotic system with hidden attractors, and the finite time synchronization of the fractional-order chaotic systems is also studied.Most of the researches mainly focus on dynamic analysis and control of integer-order chaotic systems with hidden attractors. In this paper, based on the Sprott E system, a fractional-order chaotic system is constructed by adding an appropriate constant term. The fractional-order chaotic system has only one stable equilibrium point, but it can generate various hidden attractors. Basic dynamical characteristics of the system are analyzed carefully through phase diagram,Poincare mapping and power spectrum, and the results show that the fractional-order system can present obvious chaotic characteristics. Based on bifurcation diagram of system order, it can be found that the fractional-order system can have period attractors, doubling period attractors, and chaotic attractors with various orders. Additionally, a finite time synchronization of the fractional-order chaotic system with hidden attractors is realized based on the finite time stable theorem, and the proposed controller is robust and can guarantee fast convergence. Finally, numerical simulation is carried out and the results verify the effectiveness of the proposed controller.The fractional-order chaotic system with hidden attractors has more complex and richer dynamic characteristics than integer-order chaotic systems, and chaotic range of parameters is more flexible, meanwhile the dynamics is more sensitive to system parameters. Therefore, the fractional-order chaotic system with hidden attractors can provide more key parameters and present better performance for practical applications, such as secure communication and image encryption, and it deserves to be further investigated.