深圳大学学报理工版

为满足Hopf分岔控制的实际需要和通信保密对复杂混沌系统的需求,设计一个四翼混沌吸引子和基于washout滤波器的分岔控制器.基于Lorenz系统构建新的四翼系统,通过分岔图、李雅普诺夫指数和吸引子仿真分析系统的动力学演化,确定其混沌特性.基于高维Hopf分岔理论分析系统Hopf分岔的存在性,设计了一个基于washout滤波器,由线性项和非线性项组成的Hopf分岔控制器.该控制器在不改变系统平衡点的情况下,可实现系统在预期位置产生Hopf分岔,完成对极限环幅值和稳定性的控制.仿真结果表明,极限环幅值和分岔参数的临界值是可以控制的,验证了控制器的合理性和有效性.该研究拓展了混沌系统Hopf分岔控制的多样性,在通信和图像加密领域有潜在的应用.

In order to meet the practical needs of Hopf bifurcation control and communication security, we propose a four-wing chaotic attractor and a bifurcation controller based on washout filter. We analyze the dynamic evolution of system by means of bifurcation diagram, Lyapunov exponent and attractor simulation to determine its chaotic characteristics. Based on the high-dimensional bifurcation theory, we discuss the existence of Hopf bifurcation, and design a Hopf bifurcation controller based on washout filter, which is composed of linear and nonlinear terms. Without changing the equilibrium point of system, the controller can generate the Hopf bifurcation of system at the expected position, and realize the control of amplitude and stability of the limit cycle. The simulation results show that the amplitude of the limit cycle and the critical value of bifurcation parameters can be controlled, which verifies the rationality and effectiveness of the controller. This study expands the diversity of Hopf bifurcation control in chaotic system and has potential applications in communication and image encryption.

引言
1 新四翼混沌系统
2 Hopf分岔的存在性
3 Hopf分岔控制

图1 分岔图和Lyapunov指数<br/>Fig.1 Bifurcation diagram and Lyapunov exponents

图1 分岔图和Lyapunov指数
Fig.1 Bifurcation diagram and Lyapunov exponents

图2 四翼吸引子及其相图<br/>Fig.2 Four wing attractor and the phase diagram

图2 四翼吸引子及其相图
Fig.2 Four wing attractor and the phase diagram

图3 a值为aDL和aDR时的吸引子<br/>Fig.3 Attractors for a=aDL and a=aDR

图3 a值为aDL和aDR时的吸引子
Fig.3 Attractors for a=aDL and a=aDR

图4 分岔参数与受控参数局部图<br/>Fig.4 Function diagrams of bifurcation parameter and controlled parameter

图4 分岔参数与受控参数局部图
Fig.4 Function diagrams of bifurcation parameter and controlled parameter

表1 参数a, k1和m的取值<br/>Table 1 Values of parameters of a, k1 and m

表1 参数a, k1和m的取值
Table 1 Values of parameters of a, k1 and m

图5 β2和μ2与k2关系的函数图<br/>Fig.5 Function diagram of β2, μ2 and k2

图5 β2和μ2与k2关系的函数图
Fig.5 Function diagram of β2, μ2 and k2

图6 a值为a0L和a0R时的吸引子<br/>Fig.6 Attractors with a=a0L and a=a0R

图6 a值为a0L和a0R时的吸引子
Fig.6 Attractors with a=a0L and a=a0R

图7 极限环幅值r与k2和a的关系图<br/>Fig.7 The relationship graph among the limit cycle amplitude r, k2 and a maximization step

图7 极限环幅值r与k2和a的关系图
Fig.7 The relationship graph among the limit cycle amplitude r, k2 and a maximization step

图8 不同k2和b值下的极限环<br/>Fig.8 Limit cycles with different k2 and b

图8 不同k2和b值下的极限环
Fig.8 Limit cycles with different k2 and b

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备注

引言

1 新四翼混沌系统

2 Hopf分岔的存在性

3 Hopf分岔控制

期刊信息