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两时间尺度下非光滑广义蔡氏电路系统的簇发振荡机理

李旭 张正娣 毕勤胜

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两时间尺度下非光滑广义蔡氏电路系统的簇发振荡机理

李旭, 张正娣, 毕勤胜

Mechanism of bursting oscillations in non-smooth generalized Chua’s circuit with two time scales

Li Xu, Zhang Zheng-Di, Bi Qin-Sheng
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  • 通过引入周期变化的电流源并选择适当参数, 使得周期激励频率与系统固有频率之间存在量级差距, 建立了两时间尺度即快慢耦合非光滑广义蔡氏电路模型. 基于相应的广义自治系统, 考察了其不同区域中的平衡态及其稳定性, 得到了不同分岔行为及其相应的临界条件. 同时, 利用广义Clarke导数得到的广义Jacobian矩阵, 探讨了系统轨迹穿越非光滑分界面时的各种非常规分岔模式, 进而结合广义相图, 深入分析了Fold/Fold周期簇发振荡以及Fold/Hopf周期簇 发振荡两种典型的周期簇发行为及其相应的分岔机制.
    By introducing periodically alternate current source as well as suitable values for the parameters to ensure that there exists order gap between the natural frequency and the exited frequency, a two-time scale namely, a fast-slow coupled non-smooth generalized Chua’s circuit model is established. Based on the corresponding generalized autonomous system, the stabilities of the equilibrium points in different regions are investigated, from which the critical conditions related to different types of bifurcation forms are obtained. At the same time, combining the theory of Clarke derivative, different types of non-conventional bifurcation models which may occur when the trajectory passes across the non-smooth boundaries are explored. Furthermore, with the combination of the generalized phase portraits, two typical periodic bursting phenomena namely, the Fold/Fold and Fold/Hopf periodic bursters, and their associated bifurcation mechanisms are analysed in detail.
    • 基金项目: 国家自然科学基金(批准号: 11272135, 21276115)资助的课题.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11272135, 21276115).
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    Haselbacher A, Najjar F M, Massa L, Moser R D 2010 J. Comput. Phys. 229 325

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    Merkin J H, Taylor A F 2012 Physica D 241 1336

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    Ernesto P, Dulce M, Soledad M, Jose M G, Santiago L, Julian J G 2006 Neurosci. Lett. 394 152

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    Jia Z D, Leimkuhler B 2003 Future Generation Comput. Syst. 19 415

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    Knoll D A, Chacon L, Margolin L, Mousseau V 2003 J. Comput. Phys. 185 583

    [7]

    Rinberg A, Taylor A L, Mdarder E 2013 Plos Computat. Biol. 9 e1002857

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    Strizhak P E, Kawczynski A L 1995 J. Phys. Chem. 99 10830

    [9]

    Ji Y, Bi Q S 2010 Phys. Lett. A 374 1434

    [10]

    Izhikevich E M 2000 Int. J. Bifur. Chaos 10 1171

    [11]

    Chua L O, Lin G N 1990 IEEE Trans. Circ. Syst. 37 885

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    Zhai D Q, Liu C X, Liu Y, Xu Z 2010 Acta Phys. Sin. 59 816 (in Chinese) [翟笃庆, 刘崇新, 刘尧, 许喆 2010 物理学报 59 816]

    [13]

    Chen Z Y, Zhang X F, Bi Q S 2008 Nonlin. Anal.: Real World Appl. 9 1158

    [14]

    Stouboulos I N, Miliou A N, Valaristos A P 2007 Chaos Solition. Fract. 33 1256

    [15]

    Koliopanos C L, Kyprianidis I M, Stouboulos I N 2003 Chaos Solition. Fract. 16 173

    [16]

    Yang Z M, Zhang J, Ma Y J, Bai Y L, Ma S Q 2010 Acta Phys. Sin. 59 3007 (in Chinese) [杨志民, 张洁, 马永杰, 摆玉龙, 马胜前 2010 物理学报 59 3007]

    [17]

    Binazadeh T, Shafiei M H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1071

    [18]

    Ji Y, Bi Q S 2012 Acta Phys. Sin. 61 010202 (in Chinese) [季颖, 毕勤胜 2012 物理学报 61 010202]

    [19]

    Zhang Y, Bi Q S 2011 Chin. Phys. B 20 010504-1

    [20]

    Li X H, Bi Q S 2012 Acta Phys. Sin. 61 020504 (in Chinese) [李向红, 毕勤胜 2012 物理学报 61 020504]

    [21]

    Zhang Z D, Li Y Y, Bi Q S 2013 Phys. Lett. A 377 975

  • [1]

    Shooshtari A, Pasha Zanoosi A 2010 Appl. Math. Model. 34 1918

    [2]

    Haselbacher A, Najjar F M, Massa L, Moser R D 2010 J. Comput. Phys. 229 325

    [3]

    Merkin J H, Taylor A F 2012 Physica D 241 1336

    [4]

    Ernesto P, Dulce M, Soledad M, Jose M G, Santiago L, Julian J G 2006 Neurosci. Lett. 394 152

    [5]

    Jia Z D, Leimkuhler B 2003 Future Generation Comput. Syst. 19 415

    [6]

    Knoll D A, Chacon L, Margolin L, Mousseau V 2003 J. Comput. Phys. 185 583

    [7]

    Rinberg A, Taylor A L, Mdarder E 2013 Plos Computat. Biol. 9 e1002857

    [8]

    Strizhak P E, Kawczynski A L 1995 J. Phys. Chem. 99 10830

    [9]

    Ji Y, Bi Q S 2010 Phys. Lett. A 374 1434

    [10]

    Izhikevich E M 2000 Int. J. Bifur. Chaos 10 1171

    [11]

    Chua L O, Lin G N 1990 IEEE Trans. Circ. Syst. 37 885

    [12]

    Zhai D Q, Liu C X, Liu Y, Xu Z 2010 Acta Phys. Sin. 59 816 (in Chinese) [翟笃庆, 刘崇新, 刘尧, 许喆 2010 物理学报 59 816]

    [13]

    Chen Z Y, Zhang X F, Bi Q S 2008 Nonlin. Anal.: Real World Appl. 9 1158

    [14]

    Stouboulos I N, Miliou A N, Valaristos A P 2007 Chaos Solition. Fract. 33 1256

    [15]

    Koliopanos C L, Kyprianidis I M, Stouboulos I N 2003 Chaos Solition. Fract. 16 173

    [16]

    Yang Z M, Zhang J, Ma Y J, Bai Y L, Ma S Q 2010 Acta Phys. Sin. 59 3007 (in Chinese) [杨志民, 张洁, 马永杰, 摆玉龙, 马胜前 2010 物理学报 59 3007]

    [17]

    Binazadeh T, Shafiei M H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1071

    [18]

    Ji Y, Bi Q S 2012 Acta Phys. Sin. 61 010202 (in Chinese) [季颖, 毕勤胜 2012 物理学报 61 010202]

    [19]

    Zhang Y, Bi Q S 2011 Chin. Phys. B 20 010504-1

    [20]

    Li X H, Bi Q S 2012 Acta Phys. Sin. 61 020504 (in Chinese) [李向红, 毕勤胜 2012 物理学报 61 020504]

    [21]

    Zhang Z D, Li Y Y, Bi Q S 2013 Phys. Lett. A 377 975

计量
  • 文章访问数:  2500
  • PDF下载量:  585
  • 被引次数: 0
出版历程
  • 收稿日期:  2013-07-24
  • 修回日期:  2013-08-19
  • 刊出日期:  2013-11-05

两时间尺度下非光滑广义蔡氏电路系统的簇发振荡机理

  • 1. 江苏大学理学院, 镇江 212013
    基金项目: 国家自然科学基金(批准号: 11272135, 21276115)资助的课题.

摘要: 通过引入周期变化的电流源并选择适当参数, 使得周期激励频率与系统固有频率之间存在量级差距, 建立了两时间尺度即快慢耦合非光滑广义蔡氏电路模型. 基于相应的广义自治系统, 考察了其不同区域中的平衡态及其稳定性, 得到了不同分岔行为及其相应的临界条件. 同时, 利用广义Clarke导数得到的广义Jacobian矩阵, 探讨了系统轨迹穿越非光滑分界面时的各种非常规分岔模式, 进而结合广义相图, 深入分析了Fold/Fold周期簇发振荡以及Fold/Hopf周期簇 发振荡两种典型的周期簇发行为及其相应的分岔机制.

English Abstract

参考文献 (21)

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