串联式叉型滞后簇发振荡及其动力学机制
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摘要
簇发振荡是自然界和科学技术中广泛存在的快慢动力学现象,其具有与通常的振荡显著不同的特性.根据不同的动力学机制可将其分为多种模式,例如,"点–点"型簇发振荡和"点–环"型簇发振荡等.叉型滞后簇发振荡是由延迟叉型分岔诱发的一类具有简单动力学特性的"点–点"型簇发振荡.研究以多频参数激励Duffing系统为例,旨在揭示一类与延迟叉型分岔相关的具有复杂动力学特性的簇发振荡,即串联式叉型滞后簇发振荡.考虑了一个参激频率是另一个的整倍数情形,利用"频率转换快慢分析法"得到了多频参数激励Duffing系统的快子系统和慢变量,分析了快子系统的分岔行为.研究结果表明,快子系统可以产生两个甚至多个叉型分岔点;当慢变量穿越这些叉型分岔点时,形成了两个或多个叉型滞后簇发振荡;这些簇发振荡首尾相接,最终构成了所谓的串联式叉型滞后簇发振荡.此外,分析了参数对串联式叉型滞后簇发振荡的影响.
Bursting oscillations is a spontaneous physical phenomenon existing in natural science, which has various patterns according to their dynamical regimes. For instance, bursting of point-point type means bursting patterns related to transition behaviors among different equilibrium attractors. Pitchfork-hysteresis bursting, induced by delayed pitchfork bifurcation, is a kind of point-point type bursting pattern showing simple dynamical characteristics. The present paper takes the Duffing system with multiple-frequency parametric excitations as an example in order to reveal bursting patterns, related to delayed pitchfork bifurcation, showing complex characteristics, i.e., the series-mode pitchfork-hysteresis bursting oscillations. We considered the case when one excitation frequency is an integer multiple of the other, obtained the fast subsystem and the slow variable of the Duffing system by frequency-transformation fast-slow analysis, and analyzed bifurcation behaviors of the fast subsystem. Our study shows that two or multiple pitchfork bifurcation points can be observed in the fast subsystem, and thus two or multiple pitchfork-hysteresis bursting patterns are created when the slow variable passes through these points. In particular, the pitchfork-hysteresis bursting patterns are connected in series,and as a result, the so-called series-mode pitchfork-hysteresis bursting oscillations are generated. Besides, the effects of parameters on the series-mode pitchfork-hysteresis bursting oscillations are analyzed. It is found that the damping of the system and the maximum excitation amplitude show no qualitative impact on corresponding dynamical mechanisms,while the smaller one may lead to vanish of busting oscillations. Our findings reveal the road from simple dynamical characteristics of point-point type bursting oscillation related to complex one, thereout, a complement and expansion for nowadays bursting dynamics is obtained.
Bursting oscillations is a spontaneous physical phenomenon existing in natural science, which has various patterns according to their dynamical regimes. For instance, bursting of point-point type means bursting patterns related to transition behaviors among different equilibrium attractors. Pitchfork-hysteresis bursting, induced by delayed pitchfork bifurcation, is a kind of point-point type bursting pattern showing simple dynamical characteristics. The present paper takes the Duffing system with multiple-frequency parametric excitations as an example in order to reveal bursting patterns, related to delayed pitchfork bifurcation, showing complex characteristics, i.e., the series-mode pitchfork-hysteresis bursting oscillations. We considered the case when one excitation frequency is an integer multiple of the other, obtained the fast subsystem and the slow variable of the Duffing system by frequency-transformation fast-slow analysis, and analyzed bifurcation behaviors of the fast subsystem. Our study shows that two or multiple pitchfork bifurcation points can be observed in the fast subsystem, and thus two or multiple pitchfork-hysteresis bursting patterns are created when the slow variable passes through these points. In particular, the pitchfork-hysteresis bursting patterns are connected in series,and as a result, the so-called series-mode pitchfork-hysteresis bursting oscillations are generated. Besides, the effects of parameters on the series-mode pitchfork-hysteresis bursting oscillations are analyzed. It is found that the damping of the system and the maximum excitation amplitude show no qualitative impact on corresponding dynamical mechanisms,while the smaller one may lead to vanish of busting oscillations. Our findings reveal the road from simple dynamical characteristics of point-point type bursting oscillation related to complex one, thereout, a complement and expansion for nowadays bursting dynamics is obtained.
引文
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2 Vanag VK,Zhabotinsky AM,Epstein IR.Oscillatory clusters in the periodically illuminated,spatially extended Belousov-Zhabotinsky reaction.Phys.Rev.Lett,2001,86(3):552-555
3 Winfree AT.The prehistory of the Belousov-Zhabotinsky oscillator.J.Chem.Educ,1984,61(8):661-663
4 J Rinzel.Discussion:Electrical excitability of cells,theory and experiment:Review of the Hodgkin-Huxley foundation and an update.Bull.Math.Biol,1990,52(1-2):3-23
5 Izhikevich EM.Neural excitability,spiking and bursting.Int.J.Bifurcation Chaos,2000,10(6):1171-1266
6 Rinzel J.Ordinary and Partial Differential Equations.Berlin:Springer-Verlag,1985:304-316
7 Maesschalck P De,Kutafina E,Popovi N.Sector-delayed-Hopftype mixed-mode oscillations in a prototypical three-time-scale model.Appl.Math.Comput,2016,273:337-352
8 Ruschel S,Yanchuk S.Chaotic bursting in semiconductor lasers.Chaos,2017,27:114313-114319
9 Premraj D,Suresh K,Banerjee T,et al.An experimental study of slow passage through Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator.Commun.Nonlinear Sci.Numer.Si,2016,37:212-221
10 Zhao SB,Yang LJ,Liu T,et al.Analysis of plasma oscillations by electrical detection in Nd:YAG laser welding.J Mater.Process.Tech,2017,249:479-489
11 Wilk G,Wlodarczyk Z.Temperature oscillations and sound waves in hadronic matter.Physica A,2017,486:579-586
12古华光.神经系统信息处理和异常功能的复杂动力学.力学学报,2017,49(2):410-420(Gu Huaguang.Complex dynamics of the nervous system for information processing and abnormal functions.Chinese Journal of Theoretical and Applied Mechanics,2017,49(2):410-420(in Chinese))
13 Andrew R,Esther W,Martin W,et al.Mixed mode oscillations in a conceptual climate model.Physica D,2015,292-293:70-83
14 John HG Macdonald,Guy L Larose.Two-degree-of-freedom inclined cable galloping-Part 1:General formulation and solution for perfectly tuned system.Journal of Wind Engineering and Industrial Aerodynamics,2008,96:291-307
15 Yang SC,Hong HP.Nonlinear inelastic responses of transmission tower-line system under downburst wind.Eng.Struct,2016,123:490-500
16陈章耀,陈亚光,毕勤胜.由多平衡态快子系统所诱发的簇发振荡及机理.力学学报,2015,47(4):699-706(Chen Zhangyao,Chen Yaguang,Bi Qinsheng.Bursting oscillations as well as the bifurcation mechanism induced by fast subsystem with multiple balances.Chinese Journal of Theoretical and Applied Mechanics,2015,47(4):699-706(in Chinese))
17邢雅清,陈小可,张正娣等.多平衡态下簇发振荡产生机理及吸引子结构分析.物理学报,2016,65(9):09050-1-9(Xing Yaqing,Chen Xiaoke,Zhang Zhengdi,et al.Mechanism of bursting oscillations with multiple equilibrium states and the analysis of the structures of the attractors.Acta Phys Sin,2016,65(9):09050-1-9(in Chinese))
18 Yu Y,Han XJ,Zhang C,et al.Mixed-mode oscillations in a nonlinear time delay oscillator with time varying parameters.Commun.Nonlinear Sci.Numer.Si,2017,47:23-34
19吴天一,陈小可,张正娣等.非对称型簇发振荡吸引子结构及其机理分析.物理学报,2017,66(11):35-45(Wu Tianyi,Chen Xiaoke,Zhang Zhengdi,et al.Structures of the asymmetrical bursting oscillation attractors and their bifurcation mechanisms.Acta Phys Sin,2017,66(11):35-45(in Chinese))
20 Matja P,Marko M.Different types of bursting calcium oscillations in non-excitable cells.Chaos.Soliton.Fract,2003,18:759-773
21 Han XJ,Jiang B,Bi QS.Symmetric bursting of focus-focus type in the controlled Lorenz system with two time scales.Phys.Lett.A,2009,373:3643-3649
22陈章耀,张晓芳,毕勤胜.周期激励下Hartley模型的簇发及分岔机制.力学学报,2010,42(4):765-773(Chen Zhangyao,Zhang Xiaofang,Bi Qinsheng.Bursting phenomena as well as the bifurcation mechanism in periodically excited Hartley model.Chinese Journal of Theoretical and Applied Mechanics,2010,42(4):765-773(in Chinese))
23 Han XJ,Xia FB,Ji P,et al.Hopf-bifurcation-delay-induced bursting patterns in a modified circuit system.Commun.Nonlinear Sci.Numer.Si,2016,36:517-527
24郑远广,黄承代,王在华.反馈时滞对van der Pol振子张弛振荡的影响.力学学报,2012,44(1):148-157(Zheng Yuanguang,Huang Chengdai,Wang Zaihua.Delay effect on the relaxation oscillations of a van der Pol oscillator with delayed feedback.Chinese Journal of Theoretical and Applied Mechanics,2012,44(1):148-157(in Chinese))
25 Nicolò P,Mauro S.A heterogenous Cournot duopoly with delay dynamics:Hopf bifurcations and stability switching curves.Commun.Nonlinear Sci.Numer.Si,2018,58:36-46
26 Wang HX,Zheng YH,Lu QS.Stability and bifurcation analysis in the coupled HR neurons with delayed synaptic connection.Nonlinear.Dyn,2017,88:2091-2100
27 Han XJ,Yu Y,Zhang C,et al.Turnover of hysteresis determines novel bursting in Duffing system with multiple-frequency external forcings.Int.J.Non-Linear Mech,2017,89:69-74
28 Ivana K,Michael JB.The Duffing Equation:Nonlinear Oscillators and Their Behaviour.United Kingdom:John Wiley&Sons Ltd,2011
29 Rafal R,Andrezj W,Krzysztof K,et al.Dynamics of a time delayed Duffing oscillator.Int.J.Non-Linear Mech,2014,65:98-106
30 Han XJ,Bi QS,Zhang C,et al.Delayed bifurcations to repetitive spiking and classification of delay-induced bursting.Int.J.Bifurcation Chaos,2014,24(7):1450098-1-23
31 Han XJ,Bi QS,Ji P,et al.Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies.Phys.Rev.E,2015,92:012911-1-12
32 GJM Marée.Slow passage through a pitchfork bifurcation.Slam J.Appl.Math,1996,56(3):889-918
33 Berglund N.Adiabatic dynamical systems and hysteresis.[PhDThesis].Lausanne Switzerland:Institut de Physique Th′eorique EPFL,1998
34 Han XJ,Bi QS.Bursting oscillations in Duffing’s equation with slowly changing external forcing.Commun.Nonlinear Sci.Numer.Si,2011,16:4146-4152
2 Vanag VK,Zhabotinsky AM,Epstein IR.Oscillatory clusters in the periodically illuminated,spatially extended Belousov-Zhabotinsky reaction.Phys.Rev.Lett,2001,86(3):552-555
3 Winfree AT.The prehistory of the Belousov-Zhabotinsky oscillator.J.Chem.Educ,1984,61(8):661-663
4 J Rinzel.Discussion:Electrical excitability of cells,theory and experiment:Review of the Hodgkin-Huxley foundation and an update.Bull.Math.Biol,1990,52(1-2):3-23
5 Izhikevich EM.Neural excitability,spiking and bursting.Int.J.Bifurcation Chaos,2000,10(6):1171-1266
6 Rinzel J.Ordinary and Partial Differential Equations.Berlin:Springer-Verlag,1985:304-316
7 Maesschalck P De,Kutafina E,Popovi N.Sector-delayed-Hopftype mixed-mode oscillations in a prototypical three-time-scale model.Appl.Math.Comput,2016,273:337-352
8 Ruschel S,Yanchuk S.Chaotic bursting in semiconductor lasers.Chaos,2017,27:114313-114319
9 Premraj D,Suresh K,Banerjee T,et al.An experimental study of slow passage through Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator.Commun.Nonlinear Sci.Numer.Si,2016,37:212-221
10 Zhao SB,Yang LJ,Liu T,et al.Analysis of plasma oscillations by electrical detection in Nd:YAG laser welding.J Mater.Process.Tech,2017,249:479-489
11 Wilk G,Wlodarczyk Z.Temperature oscillations and sound waves in hadronic matter.Physica A,2017,486:579-586
12古华光.神经系统信息处理和异常功能的复杂动力学.力学学报,2017,49(2):410-420(Gu Huaguang.Complex dynamics of the nervous system for information processing and abnormal functions.Chinese Journal of Theoretical and Applied Mechanics,2017,49(2):410-420(in Chinese))
13 Andrew R,Esther W,Martin W,et al.Mixed mode oscillations in a conceptual climate model.Physica D,2015,292-293:70-83
14 John HG Macdonald,Guy L Larose.Two-degree-of-freedom inclined cable galloping-Part 1:General formulation and solution for perfectly tuned system.Journal of Wind Engineering and Industrial Aerodynamics,2008,96:291-307
15 Yang SC,Hong HP.Nonlinear inelastic responses of transmission tower-line system under downburst wind.Eng.Struct,2016,123:490-500
16陈章耀,陈亚光,毕勤胜.由多平衡态快子系统所诱发的簇发振荡及机理.力学学报,2015,47(4):699-706(Chen Zhangyao,Chen Yaguang,Bi Qinsheng.Bursting oscillations as well as the bifurcation mechanism induced by fast subsystem with multiple balances.Chinese Journal of Theoretical and Applied Mechanics,2015,47(4):699-706(in Chinese))
17邢雅清,陈小可,张正娣等.多平衡态下簇发振荡产生机理及吸引子结构分析.物理学报,2016,65(9):09050-1-9(Xing Yaqing,Chen Xiaoke,Zhang Zhengdi,et al.Mechanism of bursting oscillations with multiple equilibrium states and the analysis of the structures of the attractors.Acta Phys Sin,2016,65(9):09050-1-9(in Chinese))
18 Yu Y,Han XJ,Zhang C,et al.Mixed-mode oscillations in a nonlinear time delay oscillator with time varying parameters.Commun.Nonlinear Sci.Numer.Si,2017,47:23-34
19吴天一,陈小可,张正娣等.非对称型簇发振荡吸引子结构及其机理分析.物理学报,2017,66(11):35-45(Wu Tianyi,Chen Xiaoke,Zhang Zhengdi,et al.Structures of the asymmetrical bursting oscillation attractors and their bifurcation mechanisms.Acta Phys Sin,2017,66(11):35-45(in Chinese))
20 Matja P,Marko M.Different types of bursting calcium oscillations in non-excitable cells.Chaos.Soliton.Fract,2003,18:759-773
21 Han XJ,Jiang B,Bi QS.Symmetric bursting of focus-focus type in the controlled Lorenz system with two time scales.Phys.Lett.A,2009,373:3643-3649
22陈章耀,张晓芳,毕勤胜.周期激励下Hartley模型的簇发及分岔机制.力学学报,2010,42(4):765-773(Chen Zhangyao,Zhang Xiaofang,Bi Qinsheng.Bursting phenomena as well as the bifurcation mechanism in periodically excited Hartley model.Chinese Journal of Theoretical and Applied Mechanics,2010,42(4):765-773(in Chinese))
23 Han XJ,Xia FB,Ji P,et al.Hopf-bifurcation-delay-induced bursting patterns in a modified circuit system.Commun.Nonlinear Sci.Numer.Si,2016,36:517-527
24郑远广,黄承代,王在华.反馈时滞对van der Pol振子张弛振荡的影响.力学学报,2012,44(1):148-157(Zheng Yuanguang,Huang Chengdai,Wang Zaihua.Delay effect on the relaxation oscillations of a van der Pol oscillator with delayed feedback.Chinese Journal of Theoretical and Applied Mechanics,2012,44(1):148-157(in Chinese))
25 Nicolò P,Mauro S.A heterogenous Cournot duopoly with delay dynamics:Hopf bifurcations and stability switching curves.Commun.Nonlinear Sci.Numer.Si,2018,58:36-46
26 Wang HX,Zheng YH,Lu QS.Stability and bifurcation analysis in the coupled HR neurons with delayed synaptic connection.Nonlinear.Dyn,2017,88:2091-2100
27 Han XJ,Yu Y,Zhang C,et al.Turnover of hysteresis determines novel bursting in Duffing system with multiple-frequency external forcings.Int.J.Non-Linear Mech,2017,89:69-74
28 Ivana K,Michael JB.The Duffing Equation:Nonlinear Oscillators and Their Behaviour.United Kingdom:John Wiley&Sons Ltd,2011
29 Rafal R,Andrezj W,Krzysztof K,et al.Dynamics of a time delayed Duffing oscillator.Int.J.Non-Linear Mech,2014,65:98-106
30 Han XJ,Bi QS,Zhang C,et al.Delayed bifurcations to repetitive spiking and classification of delay-induced bursting.Int.J.Bifurcation Chaos,2014,24(7):1450098-1-23
31 Han XJ,Bi QS,Ji P,et al.Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies.Phys.Rev.E,2015,92:012911-1-12
32 GJM Marée.Slow passage through a pitchfork bifurcation.Slam J.Appl.Math,1996,56(3):889-918
33 Berglund N.Adiabatic dynamical systems and hysteresis.[PhDThesis].Lausanne Switzerland:Institut de Physique Th′eorique EPFL,1998
34 Han XJ,Bi QS.Bursting oscillations in Duffing’s equation with slowly changing external forcing.Commun.Nonlinear Sci.Numer.Si,2011,16:4146-4152