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浙江大学学报(工学版)
Journal of ZheJiang University (Engineering Science)  2019, Vol. 53 Issue (8): 1552-1562    DOI: 10.3785/j.issn.1008-973X.2019.08.014
Computer and Control Engineering     
A super-wide-range parameter chaotic system with coexisting chaotic attractor
Chang-biao XU(),Zhou LI
College of Optoelectronic Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
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Abstract  

Aiming at obtaining a chaotic system with a super wide parameter range and coexisting attractors, a novel chaotic system with coexisting two-wing and four-wing attractors was constructed. There are seven items in each system state equation, of which there is only one nonlinear item. This nonlinear item is composed of the product of other two state variables. The stability of the system, the sensitivity of the system characteristics to parameter variation and the coexisting characteristics of attractors against the super-wide-range system parameter were analyzed. Results showed that the system characteristics had a strong sensitivity when parameter a was slightly changed, and the system had the characteristics of coexistence of two isolated two-wing chaotic attractors and one four-wing chaotic attractor when only the initial value was changed, and the system had coexisting chaotic attractors namely two isolated two-wing chaotic attractors and one four-wing chaotic attractor for parameter d∈(0, 2×104]. In addition, a hardware circuit of the system was designed and circuit simulation was conducted by Multisim. The existence of coexisting attractors under the wide variation range of parameters was further verified.

Key words： chaotic system      super-wide range      multiple coexisting attractors      sensitivity      circuit implementation     
Received: 05 July 2018      Published: 13 August 2019
CLC:  TP 11  
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Chang-biao XU
Zhou LI
Cite this article:

Chang-biao XU,Zhou LI. A super-wide-range parameter chaotic system with coexisting chaotic attractor. Journal of ZheJiang University (Engineering Science), 2019, 53(8): 1552-1562.

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http://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2019.08.014     OR     http://www.zjujournals.com/eng/Y2019/V53/I8/1552


具有共存混沌吸引子的超大范围参数混沌系统

为了使参数在超大范围内变化时系统均具有共存吸引子，构建新型的双翼与四翼吸引子共存的混沌系统. 系统的状态方程共有7项，在每个状态方程中只有1个非线性项，且此非线性项是由另外2个状态变量的乘积组成的. 分析系统的稳定性、系统特性对参数变化的敏感性、系统参数在超大范围内变化时吸引子的共存特性等. 研究结果表明，在参数α作微小变化时，系统特性具有较强的敏感性;当仅改变初始值的大小时，系统具有2个孤立双翼混沌吸引子与1个四翼混沌吸引子共存的特性；当参数d∈(0, 2×104]时，系统同样具有混沌吸引子，且均具有共存的2个孤立双翼混沌吸引子与1个四翼混沌吸引子. 此外,设计系统的硬件电路，利用Multisim进行电路仿真，进一步验证参数在超大范围内变化时系统中共存吸引子的存在性.


关键词： 混沌系统,  超大范围,  多重共存吸引子,  敏感性,  电路实现 
Fig.1 Phase diagrams of constructed novel chaotic system
平衡点 特征根 稳定性
S0 λ1=?3，λ2=?1为负实数；λ3=2为正实数 不稳定鞍点
S1 λ1=?3.114为负实数；λ2=0.557?2.239i，λ3=0.557+2.239i，均有正实部 不稳定的鞍焦点
S2、S3 λ1=?4.848为负实数；λ2=1.424?2.632i，λ3=1.424+2.632i，均有正实部 不稳定的鞍焦点
S4 λ1=?5.102为负实数；λ2=1.051?4.179i，λ3=1.051+4.179i，均有正实部 不稳定的鞍焦点
Tab.1 Stability of equilibrium points in novel chaotic system
Fig.2 Time domain waveform of chaotic system
Fig.3 Poincaré section of chaotic system
Fig.4 Lyapunov exponent spectrum of chaotic system
Fig.5 Bifurcation diagrams of system variable with change of parameters
Fig.6 x-z phase diagrams for different values of parameter a
参数a Lyapunov指数
a = ?1.5 0.473 6, 0.007 2, ? 2.480 9
a = ?0.95 0.296 9, ?0.007 1, ?2.289 8
a = 0 0.004 2, ?0.992 2, ?1.012 0
a = 0.95 0.316 3, ?0.003 1, ?2.313 2
a = 1.00 0.395 1, 0.003 7, ?2.398 8
Tab.2 Lyapunov exponents with different values of parameter a
Fig.7 x-y phase diagrams and bifurcation diagrams corresponding to different initial values
Fig.8 x-z phase diagrams and bifurcation diagrams corresponding to different initial values
Fig.9 Phase diagrams and bifurcation diagrams corresponding to different initial values for d = 10
Fig.12 Phase diagrams and bifurcation diagrams corresponding to different initial values for d = 10 000
Fig.10 Phase diagrams and bifurcation diagrams corresponding to different initial values for d = 100
Fig.11 Phase diagrams and bifurcation diagrams corresponding to different initial values for d = 1 000
Fig.13 Circuit schematic diagram of chaotic system
Fig.14 Circuit experiment results of chaotic system
Fig.15 Phase diagrams for d = 10, 100, 300
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