Bursting Oscillations and Bifurcation Mechanism in Memristor-based Shimizu–Morioka System with Multi-frequency Slow Excitations
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摘要: 为了研究忆阻系统的簇发振荡及其形成机理，该文在Shimizu-Morioka(S-M)系统的基础上引入忆阻器件和两个慢变化的周期激励项，建立了一种多时间尺度的忆阻型S-M系统。首先研究了单一激励下S-M系统的簇发行为及分岔机制，得到一种对称型“sub-Hopf/sub-Hopf”簇发模式。然后借助De Moivre公式将多频激励系统转化为单频激励系统，结合快慢分析法重点分析了附加激励幅度对“sub-Hopf/sub-Hopf”簇发模式的影响。对应于不同附加激励幅度该文发现了两种新的簇发模式，即扭曲型“sub-Hopf/sub-Hopf”簇发和嵌套级联型sub-Hopf/sub-Hopf”簇发。借助时序图、分岔图和转换相图分析了相应的簇发机制。最后，采用Multisim软件搭建电路模型并进行仿真实验，得到的实验结果与理论分析结果相吻合，从而实验证明了忆阻型S-M系统的簇发模式。Abstract: In order to study the bursting oscillations and its formation mechanism of memristor-based system, a multi-timescale memristor-based S-M system is established by introducing a memristor device and two slowly changing periodic excitations into the Shimizu-Morioka (S-M) system. Firstly, the bursting behavior and bifurcation mechanism of S-M system under single excitation are studied, and a symmetric bursting pattern of “sub-Hopf/sub-Hopf” is obtained. Then the multi-frequency excitation system is transformed into single frequency excitation system by using De Moivre formula, and the influence of additional excitation amplitude and frequency on “sub Hopf / sub Hopf” bursting mode is analyzed by using the fast-slow analysis method. As a result, two new bursting patterns named as twisted “sub-Hopf/sub-Hopf” bursting and nested “sub-Hopf/sub-Hopf” are found under different amplitudes of the additional excitation. The corresponding bursting mechanisms are analyzed with time history diagram, bifurcation diagram and transformation phase diagram. Finally, Multisim simulation results, which are in good agreement with the numerical simulation results, are provided to verify the validity of the study.
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