Journal of Zhejiang University-SCIENCE A

, Volume 19, Issue 12, pp 889–903 | Cite as

Synchronization stability between initial-dependent oscillators with periodical and chaotic oscillation

  • Fu-qiang Wu
  • Jun Ma
  • Guo-dong Ren


The selection of periodical or chaotic attractors becomes initial-dependent in that setting different initial values can trigger a different profile of attractors in a dynamical system with memory by adding a nonlinear term such as z2y in the Rössler system. The memory effect means that the outputs are very dependent on the initial value for variable z, e.g. magnetic flux for a memristor. In this study, standard nonlinear analyses, including phase portrait, bifurcation analysis, and Lyapunov exponent analysis were carried out. Synchronization between two coupled oscillators and a network was investigated by resetting initial states. A statistical synchronization factor was calculated to find the dependence of synchronization on the coupling intensity when different initial values were selected. Our results show that the dynamics of the attractor depends on the selection of the initial value for one variable z. In the case of coupling between two oscillators, appropriate initial values are selected to trigger two different nonlinear oscillators (periodical and chaotic). Results show that complete synchronization between periodical oscillators, chaotic oscillators, and periodical and chaotic oscillators can be realized by applying an appropriate unidirectional coupling intensity. In particular, two periodical oscillators can be coupled bidirectionally to reach chaotic synchronization so that periodical oscillation is modulated to become chaotic. When the memory effect is considered on some nodes of a chain network, enhancement of memory function can decrease the synchronization, while a small region for intensity of memory function can contribute to the synchronization of the network. Finally, dependence of attractor formation on the initial setting was verified on the field programmable gate array (FPGA) circuit in digital signal processing (DSP) builder block under Matlab/Simulink.

Key words

Synchronization Bifurcation Synchronization factor Field programmable gate array (FPGA) 






1. 两个周期振子耦合后达到混沌同步;2. 周期振子和混沌振子耦合后达到周期性振荡同步。


1. 通过分岔分析,研究振荡模态和初始值选择之间的关系(图2、6 和8);2. 通过数值计算,研究两个周期振子在耦合下的混沌同步关系(图 7);3. 通过计算同步因子和斑图,分析同步一致 性对耦合强度与记忆函数增益的依赖程度(图9 和10);4. 通过现场可编程门阵列验证动力系统 模态对初始值的依赖程度(图11 和12)。


1. 具有记忆函数的非线性振子的动力学行为(如吸引子)在参数固定的情况下与初始值选取有 关。2. 不同类型振子的耦合可以达到多样同步行为;周期振子耦合达到混沌同步;周期振子耦合 混沌振子可以抑制混沌。3. 包含记忆函数的振子 网络耦合同步非常困难。


同步 分岔 同步因子 现场可编程门阵列 

CLC number

O59 TN710 


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Copyright information

© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsLanzhou University of TechnologyLanzhouChina
  2. 2.School of ScienceChongqing University of Posts and TelecommunicationsChongqingChina
  3. 3.NAAM-Research Group, Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia

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