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非对称型簇发振荡吸引子结构及其机理分析

吴天一 陈小可 张正娣 张晓芳 毕勤胜

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非对称型簇发振荡吸引子结构及其机理分析

吴天一, 陈小可, 张正娣, 张晓芳, 毕勤胜

Structures of the asymmetrical bursting oscillation attractors and their bifurcation mechanisms

Wu Tian-Yi, Chen Xiao-Ke, Zhang Zheng-Di, Zhang Xiao-Fang, Bi Qin-Sheng
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  • 旨在揭示频域不同尺度耦合时非对称动力系统簇发振荡的特点及其分岔机理,并进一步揭示快子系统多平衡点共存导致的不同簇发模式及其产生原因.以经典的蔡氏振子为例,通过引入非对称控制项及周期变化的电流源,选取适当参数,构建存在频域两尺度耦合的非对称动力系统模型.当周期激励频率远小于系统的固有频率时,将整个周期激励项视为慢变参数,得到随慢变参数变化的快子系统平衡曲线及其不同的分岔点以及分岔行为.重点分析了三种不同周期激励幅值下典型的非对称簇发振荡及吸引子结构,揭示其相应的产生机理.指出外激励幅值的变化不仅会引起不同稳定平衡点吸引域的变化,也会使得慢变量穿越不同分岔点的时间间隔发生变化,导致系统产生不同形式的簇发振荡.
    The main purpose of this study is to investigate the characteristics as well as the bifurcation mechanisms of the bursting oscillations in the asymmetrical dynamical system with two scales in the frequency domain. Since the slow-fast Hodgkin-Huxley model was established to successfully reproduce the activities of neuron, the complicated dynamics of the system with multiple time scales has become a hot research topic due to the wide engineering background. The dynamical system with multiple scales often presents periodic oscillations coupled by large-amplitude oscillations at spiking states and small-amplitude oscillations at quiescent states, which are connected by bifurcations. Up to now, most of the reports concentrate on bursting oscillations in the symmetric systems, in which there exists only one form of spiking oscillations and quiescence, respectively. Here we explore some typical forms of bursting behavior in an asymmetrical dynamical system with periodic excitation, in which there exists an order gap between the exciting frequency and the natural frequency. As an example, based on the typical Chua's oscillator, by introducing an asymmetrical controller and a periodically changed current source, and choosing suitable parameter values, we establish an asymmetrical dynamical system with two scales in the frequency domain. Since the exciting frequency is much smaller than the natural frequency, the whole periodic exciting term can be regarded as a slowly-varying parameter, leading to the fast subsystem in autonomous form. Since all the equilibrium curves and relevant bifurcations are presented in the form related to the slowly-varying parameter, the transformed phase portraits describing the evolution relationship between the state variables and the slowly-varying parameter are employed to account for the mechanism of the bursting oscillations. With the variation of the slowly-varying parameter, different equilibrium states and relevant bifurcations in the fast subsystem are presented. It is found that for different parameter values, multiple balance curves of the fast subsystem may coexist, which affect the structure of the bursting attractor. For the other parameters fixed to certain values, the balance curve with the variation of the slowly-varying parameter is presented. Three typical cases with different exciting amplitudes are considered, corresponding to different situations of coexistence of equilibrium states in the fast subsystem. In the first case, there exist at most three stable equilibrium points in the fast subsystem. Bursting attractor that oscillates around the three points can be observed, in which fold and Hopf bifurcations lead to the alternations between spiking states and quiescent states, while in the second case, saddle on the limit cycle bifurcation may cause the repetitive spiking oscillations to jump to the equilibrium curve. In the third case with relatively large exciting amplitude, only two equilibrium curves may involve the bursting oscillations, in which fold bifurcations lead to the alternation between the quiescent states and spiking states. Unlike the structures of bursting oscillations in the symmetric system, different forms of asymmetrical bursting oscillations with different periodic exciting amplitudes can be observed, the mechanisms of which are presented. It is pointed out that the change of the external exciting amplitude, does not only cause the variation of the attracting basins corresponding to different stable equilibrium branches, but also leads to the change of the temporal intervals when the trajectory passes different bifurcation points, respectively, which results in different patterns of bursting oscillations. Furthermore, since the slowly-varying parameter determined by the whole exciting term changes between two extreme values determined by the amplitude, the trajectory of the bursting oscillations of the transformed phase portrait returns at the two extreme values. The properties of equilibrium branches between the two extreme values determine the forms of the moving attractors.
      通信作者: 毕勤胜, qbi@ujs.edu.cn
    • 基金项目: 国家自然科学基金(批准号:11472115,11472116)和国家自然科学基金重点项目(批准号:11632008)资助的课题.
      Corresponding author: Bi Qin-Sheng, qbi@ujs.edu.cn
    • Funds: Supported by the National Natural Science Foundation of China (Grant Nos. 11472115, 11472116) and the Major National Natural Science Foundation of China (Grant No. 11632008).
    [1]

    Cardin P T, de Moraes J R, da Silva P R 2015 J. Math. Anal. Appl. 423 1166

    [2]

    Hodgkin A L, Huxley A F 1990 Bull. Math. Biol. 52 25

    [3]

    Sánchez A D, G.Izús G, Erba M G, Deza R R 2014 Phys. Lett. A 378 1579

    [4]

    Rakaric Z, Kovacic I 2016 Mech. Syst. Signal Process. 81 35

    [5]

    Chumakov G A, Chumakova N A 2003 Chem. Eng. J. 91 151

    [6]

    Jia F L, Xu W, Li H N, Hou L Q 2013 Acta Phys. Sin. 62 100503 (in Chinese) [贾飞蕾, 徐伟, 李恒年, 侯黎强 2013 物理学报 62 100503]

    [7]

    Yang S C, Hong H P 2016 Eng. Struct. 123 490

    [8]

    Rakaric Z, Kovacic I 2016 Mech. Syst. Signal Process. 81 35

    [9]

    Li X H, Hou J Y 2016 Int. J. Non-Linear Mech. 81 165

    [10]

    Yu B S, Jin D P, Pang Z J 2014 Sci. Sin. Phys. Mech. Astron. 44 858 (in Chinese) [余本嵩, 金栋平, 庞兆君 2014 中国科学: 物理学 力学 天文学 44 858]

    [11]

    Bi Q S 2012 Sci. China Ser. E10 2820

    [12]

    Kim S, Lim W 2016 Neural Networks 79 53

    [13]

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

    [14]

    Izhikevich E M 2003 Trends Neurosci. 26 161

    [15]

    Shimizu K, Saito Y, Sekikawa M, Inaba N 2012 Physica D 241 1518

    [16]

    Shilnikov A, Kolomiets M 2008 Int. J. Bifurcation Chaos 18 2141

    [17]

    Han X J, Xia F B, Ji P, Bi Q S, Kurths J 2016 Commun. Nonlinear Sci. Numer. Si. 36 517

    [18]

    Bi Q S, Zhang R, Zhang Z D 2014 Appl. Math. Comput. 243 482

    [19]

    Yu Y, Zhang Z D, Bi Q S, Gao Y 2016 Appl. Math. Model 40 1816

    [20]

    Zhang X F, Wu L, Bi Q S 2016 Chin. Phys. B 25 070501

    [21]

    Xing Y Q, Chen X K, Zhang Z D, Bi Q S 2016 Acta Phys. Sin. 65 090501 (in Chinese) [邢雅清, 陈小可, 张正娣, 毕勤胜 2016 物理学报 65 090501]

    [22]

    Zheng S, Han X J, Bi Q S 2011 Commun. Nonlinear Sci. Numer. Si. 16 1998

    [23]

    Vo T, Kramer M A, Kaper T J 2016 Phys. Rev. Lett. 117 268101

    [24]

    Zvonko R, Ivana K 2016 Mech. Syst. Signal Process. 81 35

    [25]

    Milicevic K, Nyarko E K, Biondic I 2016 Nonlinear Dyn. 81 51

    [26]

    Srinivasan K, Chandrasekar V K, Pradeep R G 2016 Commun. Nonlinear Sci. Numer. Si. 39 156

  • [1]

    Cardin P T, de Moraes J R, da Silva P R 2015 J. Math. Anal. Appl. 423 1166

    [2]

    Hodgkin A L, Huxley A F 1990 Bull. Math. Biol. 52 25

    [3]

    Sánchez A D, G.Izús G, Erba M G, Deza R R 2014 Phys. Lett. A 378 1579

    [4]

    Rakaric Z, Kovacic I 2016 Mech. Syst. Signal Process. 81 35

    [5]

    Chumakov G A, Chumakova N A 2003 Chem. Eng. J. 91 151

    [6]

    Jia F L, Xu W, Li H N, Hou L Q 2013 Acta Phys. Sin. 62 100503 (in Chinese) [贾飞蕾, 徐伟, 李恒年, 侯黎强 2013 物理学报 62 100503]

    [7]

    Yang S C, Hong H P 2016 Eng. Struct. 123 490

    [8]

    Rakaric Z, Kovacic I 2016 Mech. Syst. Signal Process. 81 35

    [9]

    Li X H, Hou J Y 2016 Int. J. Non-Linear Mech. 81 165

    [10]

    Yu B S, Jin D P, Pang Z J 2014 Sci. Sin. Phys. Mech. Astron. 44 858 (in Chinese) [余本嵩, 金栋平, 庞兆君 2014 中国科学: 物理学 力学 天文学 44 858]

    [11]

    Bi Q S 2012 Sci. China Ser. E10 2820

    [12]

    Kim S, Lim W 2016 Neural Networks 79 53

    [13]

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

    [14]

    Izhikevich E M 2003 Trends Neurosci. 26 161

    [15]

    Shimizu K, Saito Y, Sekikawa M, Inaba N 2012 Physica D 241 1518

    [16]

    Shilnikov A, Kolomiets M 2008 Int. J. Bifurcation Chaos 18 2141

    [17]

    Han X J, Xia F B, Ji P, Bi Q S, Kurths J 2016 Commun. Nonlinear Sci. Numer. Si. 36 517

    [18]

    Bi Q S, Zhang R, Zhang Z D 2014 Appl. Math. Comput. 243 482

    [19]

    Yu Y, Zhang Z D, Bi Q S, Gao Y 2016 Appl. Math. Model 40 1816

    [20]

    Zhang X F, Wu L, Bi Q S 2016 Chin. Phys. B 25 070501

    [21]

    Xing Y Q, Chen X K, Zhang Z D, Bi Q S 2016 Acta Phys. Sin. 65 090501 (in Chinese) [邢雅清, 陈小可, 张正娣, 毕勤胜 2016 物理学报 65 090501]

    [22]

    Zheng S, Han X J, Bi Q S 2011 Commun. Nonlinear Sci. Numer. Si. 16 1998

    [23]

    Vo T, Kramer M A, Kaper T J 2016 Phys. Rev. Lett. 117 268101

    [24]

    Zvonko R, Ivana K 2016 Mech. Syst. Signal Process. 81 35

    [25]

    Milicevic K, Nyarko E K, Biondic I 2016 Nonlinear Dyn. 81 51

    [26]

    Srinivasan K, Chandrasekar V K, Pradeep R G 2016 Commun. Nonlinear Sci. Numer. Si. 39 156

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出版历程
  • 收稿日期:  2016-12-09
  • 修回日期:  2017-01-16
  • 刊出日期:  2017-06-05

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