正负双向脉冲式爆炸及其诱导的簇发振荡
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摘要
簇发振荡普遍存在.探索通向簇发振荡的可能路径是簇发研究的热点问题之一."脉冲式爆炸(pulsedshaped explosion, PSE)"是一种最近被报道的可以诱发簇发振荡的新机制,其特征为平衡点和极限环表现出了与参数变化相关的脉冲式急剧量变. PSE会导致系统轨线急剧跃迁,从而诱发典型的簇发振荡.然而,目前报道的PSE中仅含有"单向的尖峰",未发现"双向的尖峰",且由其诱发的簇发振荡仅含单向的振荡簇.本文以多频激励Rayleigh系统为例,旨在揭示PSE的不同表现形式以及与此相关的簇发动力学.利用频率转换快慢分析法得到了Rayleigh系统的快子系统和慢变量.针对快子系统的分析表明,PSE表现出了较为复杂的动力学特性,其特征是PSE包含了正负双向两个不同的尖峰,此即所谓的正负双向PSE.其急剧量变行为,导致了系统轨线在单个振荡周期内出现正向和负向的多次跃迁,由此得到了由正负双向PSE所诱发的簇发振荡.根据吸引子类型分别揭示了点–点型和环–环型两类簇发振荡模式的产生机制.本文的研究给出了PSE的不同表现形式,丰富了多时间尺度下的簇发振荡的诱发机制.
Bursting oscillations, characterized by the alternation between large-amplitude oscillations and small-amplitude oscillations, are complex behaviors of dynamical systems with multiple time scales and have become one of the hot subjects in nonlinear science. Up to now, various underlying mechanisms of bursting oscillations as well as the classifications have been investigated intensively. Recently, a sharp transition behavior, called the "pulsed-shaped explosion(PSE)",was uncovered based on nonlinear oscillators of Rayleigh's type. PSE is characterized by pulse-shaped sharp quantitative changes appearing in the branches of equilibrium point and limit cycle. However, the previous work related to the PSE merely focused on the sharp transitions of unidirectional PSE, and more complex forms of PSE which may lead to more complicated bursting patterns need to be further investigated. Taking a parametrically and externally excited Rayleigh system as an example, we reveal different expression of PSE as well as bursting patterns induced by it. According to the frequency relationship between the two slow excitations, the fast subsystem and the slow variable are obtained by means of frequency-transformation fast-slow analysis. Bifurcation behaviors of the fast subsystem show that, there exist two originally disunited branches of equilibrium point and limit cycle, which extend steeply in different directions and inosculate as a integral structure according to the variation of system parameters. This inosculated integral structure inherits the "steep" properties in different directions from the originally disunited branches, and PSE is thus generated.Unlike the PSE phenomena studied in the previous works, the PSE reported here contains two different peaks in positive and negative directions, which can be named as "positive and negative PSE". Note that the positive and negative PSE essentially complicates bursting dynamics and plays a critical role in bursting. On the other hand, only with the properly chosen parameters could it be created. Based on this, two different types of bursting, i.e., point-point type and cycle-cycle type, are obtained. Subsequently, the transformed phase portraits are introduced to explore dynamical mechanisms of the bursting patterns. We show that, with the variation of the slow parameter, the trajectory may undergo sharp transitions between the rest and active states by positive and negative PSE, and therein lies the generation of bursting. Our results demonstrate the diversity of PSE and give a complement to the underlying mechanisms of bursting.
Bursting oscillations, characterized by the alternation between large-amplitude oscillations and small-amplitude oscillations, are complex behaviors of dynamical systems with multiple time scales and have become one of the hot subjects in nonlinear science. Up to now, various underlying mechanisms of bursting oscillations as well as the classifications have been investigated intensively. Recently, a sharp transition behavior, called the "pulsed-shaped explosion(PSE)",was uncovered based on nonlinear oscillators of Rayleigh's type. PSE is characterized by pulse-shaped sharp quantitative changes appearing in the branches of equilibrium point and limit cycle. However, the previous work related to the PSE merely focused on the sharp transitions of unidirectional PSE, and more complex forms of PSE which may lead to more complicated bursting patterns need to be further investigated. Taking a parametrically and externally excited Rayleigh system as an example, we reveal different expression of PSE as well as bursting patterns induced by it. According to the frequency relationship between the two slow excitations, the fast subsystem and the slow variable are obtained by means of frequency-transformation fast-slow analysis. Bifurcation behaviors of the fast subsystem show that, there exist two originally disunited branches of equilibrium point and limit cycle, which extend steeply in different directions and inosculate as a integral structure according to the variation of system parameters. This inosculated integral structure inherits the "steep" properties in different directions from the originally disunited branches, and PSE is thus generated.Unlike the PSE phenomena studied in the previous works, the PSE reported here contains two different peaks in positive and negative directions, which can be named as "positive and negative PSE". Note that the positive and negative PSE essentially complicates bursting dynamics and plays a critical role in bursting. On the other hand, only with the properly chosen parameters could it be created. Based on this, two different types of bursting, i.e., point-point type and cycle-cycle type, are obtained. Subsequently, the transformed phase portraits are introduced to explore dynamical mechanisms of the bursting patterns. We show that, with the variation of the slow parameter, the trajectory may undergo sharp transitions between the rest and active states by positive and negative PSE, and therein lies the generation of bursting. Our results demonstrate the diversity of PSE and give a complement to the underlying mechanisms of bursting.
引文
1 Wang Y,Zhang L.Existence of asymptotically stable periodic solutions of a Rayleigh type equation.Nonlinear Analysis,2009,71(5):1728-1735
2 Han XJ,Xia FB,Zhang C,et al.Origin of mixed-mode oscillations through speed escape of attractors in a Rayleigh equation with multiple-frequency excitations.Nonlinear Dynamics,2017,88(4):2693-2703
3 Warminski J.Nonlinear normal modes of a self-excited system driven by parametric and external excitations.Nonlinear Dynamics,2010,61(4):677-689
4 Bikdash M,Balachandran B,Navfeh A.Melnikov analysis for a ship with a general roll-damping model.Nonlinear Dynamics,1994,6(1):101-124
5 Desai YM,Yu P,Popplewell N,et al.Finite element modelling of transmission line galloping.Computers&Structures,1995,57(3):407-420
6 Felix JLP,Balthazar JM,Brasil RMLRF.Comments on nonlinear dynamics of a non-ideal Duffing-Rayleigh oscillator:Numerical and analytical approaches.Journal of Sound&Vibration,2009,319(3):1136-1149
7 Warminski J,Balthazar JM.Vibrations of a parametrically and selfexcited system with ideal and non-ideal energy sources.Journal of the Brazilian Society of Mechanical Sciences&Engineering,2003,25(4):413-420
8 Kumar P,Kumar A,Erlicher S.A modified hybrid van der PolDuffing-Rayleigh oscillator for modelling the lateral walking force on a rigid floor.Physica D,2017,358:1-14
9 Tabejieu LMA,Nbendjo BRN,Filatrella G,et al.Amplitude stochastic response of Rayleigh beams to randomly moving loads.Nonlinear Dynamics,2017,89(2):925-937
10黄显高,徐健学,何岱海等.利用小波多尺度分解算法实现混沌系统的噪声减缩.物理学报,1999,48(10):1810-1817(Huang Xiangao,Xu Jianxue,He Daihai,et al.Reduction of noise in chaotic systems by wavelet multiscaling decomposition algorithm.Acta Phys.Sin,1999,48(10):1810-1817(in Chinese))
11王帅,于文浩,陈巨辉等.鼓泡流化床中流动特性的多尺度数值模拟.力学学报,2016,48(3):585-592(Wang Shuai,Yu Wenhao,Chen Juhui,et al.Multi-scale simulation on hydrodynamic characteristics in bubbling fluidized bed.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):585-592(in Chinese))
12贾宏涛,许春晓,崔桂香.槽道湍流近壁区多尺度输运特性研究.力学学报,2007,39(2):181-187(Jia Hongtao,Xu Chunxiao,Cui Guixiang.Multi-scale energy transfer in near-wall region of turbulentchannel flow.Chinese Journal of Theoretical and Applied Mechanics,2007,39(2):181-187(in Chinese))
13 Demongeot J,Bezy-Wendling J,Mattes J,et al.Multiscale modeling and imaging:the challenges of biocomplexity.Proceedings of the IEEE,2003,91(10):1723-1737
14 Savino GV,Formigli CM.Nonlinear electronic circuit with neuron like bursting and spiking dynamics.Biosystems,2009,97(1):9-14
15 Lu QS,Gu HG,Yang ZQ,et al.Dynamics of firing patterns,synchronization and resonances in neuronal electrical activities:Experiments and analysis.Acta Mech.Sin,2008,24(6):593-628
16 Sriram K,Gopinathan MS.Effects of delayed linear electrical perturbation of the Belousov-Zhabotinsky reaction:A case of complex mixed mode oscillations in a batch reactor.React.Kinet.Catal.Leu,2003,79(2):341-349
17 Wu HG,Bao BC,Liu Z,et al.Chaotic and period bursting phenomena in a memristive Wien-bridge oscillator.Nonlinear Dynamics,2016,83(1-2):893-903
18 Wang HX,Zheng YH,Lu QS.Stability and bifurcation analysis in the coupled HR neurons with delayed synaptic connection.Nonlinear Dynamics,2017,88(3):2091-2100
19陈振阳,韩修静,毕勤胜.一类二维非自治离散系统中的复杂簇发振荡结构.力学学报,2017,49(1):165-174(Chen Zhenyang,Han Xiujing,Bi Qinsheng.Complex bursting oscillation structures in a two-dimensional non-autonomous discrete system.Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):165-174(in Chinese))
20陈振阳,韩修静,毕勤胜.离散达芬映射中由边界激变所诱发的复杂的张弛振荡.力学学报,2017,49(6):1380-1389(Chen Zhenyang,Han Xiujing,Bi Qinsheng.Complex relaxation oscillation triggered by boundary crisis in the discrete Duffing map.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1380-1389(in Chinese))
21曲子芳,张正娣,彭淼等.双频激励下Filippov系统的非光滑簇发振荡机理.力学学报,2018,50(5):1145-1155(Qu Zifang,Zhang Zhengdi,Peng Miao,et al.Non-Smooth bursting oscillation mechanisms in a Filippov-type system with multiple periodic excitations.Chinese Journal of Theoretical and Applied Mechanics,2018,50(5):1145-1155(in Chinese))
22 Koper MTM.Bifurcations of mixed-mode oscillations in a threevariable autonomous van der Pol-Duffing model with a cross-shaped phase diagram.Physica D,1995,80(1):72-94
23毕勤胜,陈章耀,朱玉萍等.参数激励耦合系统的复杂动力学行为分析.力学学报,2003,35(3):367-372(Bi Qinsheng,Chen Zhangyao,Zhu Yuping,et al.Dynamical analysis of coupled oscillators with parametrical excitation.Chinese Journal of Theoretical and Applied Mechanics,2003,35(3):367-372(in Chinese))
24 Curtu R.Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network.Physica D,2010,239(9):504-514
25 Marino F,Ciszak M,Abdalah SF,et al.Mixed-mode oscillations via canard explosions in light-emitting diodes with optoelectronic feedback.Phys.Rev.E,2011,84(4 Pt 2):047201
26 Hou JY,Li XH,Zuo DW,et al.Bursting and delay behavior in the Belousov-Zhabotinsky reaction with external excitation.Eur.Phys.J.Plus,2017,132(6):283
27 Yu Y,Zhang C,Han XJ.Routes to bursting in active control system with multiple time delays.Nonlinear Dynamics,2017,88(3):2241-2254
28 Han XJ,Bi QS,Kurths J.Route to bursting via pulse-shaped explosion.Phys.Rev.E,2018,98(1):010201
29 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(1):012911
30 Rinzel J.Bursting oscillations in an excitable membrane model//Sleeman BD,Jarvis RJ.Ordinary and Partial Differential Equations.Berlin:Springer-Verlag,1985:304-316
31 Izhikevich EM.Neural excitability,spiking and bursting.Int.J.Bifurcation Chaos,2000,10(6):1171-1266
32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,2015,4(4):699-706(Chen Zhangyao,ChenYaguang,Bi Qinsheng.Bursting oscillation as well as the bifurcation mechanism induced by fast subsystem with multiple balances.Chinese Journal of Theoretical and Applied Mechanics,2015,4(4):699-706(in Chinese))
33 Stankevich N,Mosekilde E.Coexistence between silent and bursting states in a biophysical Hodgkin-Huxley-type of model.Chaos,2017,27(12):123101
34夏雨,毕勤胜,罗超等.双频1:2激励下修正蔡氏振子两尺度耦合行为.力学学报,2018,50(2):362-372(Xia Yu,Bi Qinsheng,Luo Chao,et al.Behaviors of modified Chua’s oscillator two time scales under two excitatoins with frequency ratio at 1:2.Chinese Journal of Theoretical and Applied Mechanics,2018,50(2):362-372(in Chinese))
35 Yu Y,Zhao M,Zhang ZD.Novel bursting patterns in a van der PolDuffing oscillator with slow varying external force.Mech.Syst.Signal Proc,2017,93:164-174
2 Han XJ,Xia FB,Zhang C,et al.Origin of mixed-mode oscillations through speed escape of attractors in a Rayleigh equation with multiple-frequency excitations.Nonlinear Dynamics,2017,88(4):2693-2703
3 Warminski J.Nonlinear normal modes of a self-excited system driven by parametric and external excitations.Nonlinear Dynamics,2010,61(4):677-689
4 Bikdash M,Balachandran B,Navfeh A.Melnikov analysis for a ship with a general roll-damping model.Nonlinear Dynamics,1994,6(1):101-124
5 Desai YM,Yu P,Popplewell N,et al.Finite element modelling of transmission line galloping.Computers&Structures,1995,57(3):407-420
6 Felix JLP,Balthazar JM,Brasil RMLRF.Comments on nonlinear dynamics of a non-ideal Duffing-Rayleigh oscillator:Numerical and analytical approaches.Journal of Sound&Vibration,2009,319(3):1136-1149
7 Warminski J,Balthazar JM.Vibrations of a parametrically and selfexcited system with ideal and non-ideal energy sources.Journal of the Brazilian Society of Mechanical Sciences&Engineering,2003,25(4):413-420
8 Kumar P,Kumar A,Erlicher S.A modified hybrid van der PolDuffing-Rayleigh oscillator for modelling the lateral walking force on a rigid floor.Physica D,2017,358:1-14
9 Tabejieu LMA,Nbendjo BRN,Filatrella G,et al.Amplitude stochastic response of Rayleigh beams to randomly moving loads.Nonlinear Dynamics,2017,89(2):925-937
10黄显高,徐健学,何岱海等.利用小波多尺度分解算法实现混沌系统的噪声减缩.物理学报,1999,48(10):1810-1817(Huang Xiangao,Xu Jianxue,He Daihai,et al.Reduction of noise in chaotic systems by wavelet multiscaling decomposition algorithm.Acta Phys.Sin,1999,48(10):1810-1817(in Chinese))
11王帅,于文浩,陈巨辉等.鼓泡流化床中流动特性的多尺度数值模拟.力学学报,2016,48(3):585-592(Wang Shuai,Yu Wenhao,Chen Juhui,et al.Multi-scale simulation on hydrodynamic characteristics in bubbling fluidized bed.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):585-592(in Chinese))
12贾宏涛,许春晓,崔桂香.槽道湍流近壁区多尺度输运特性研究.力学学报,2007,39(2):181-187(Jia Hongtao,Xu Chunxiao,Cui Guixiang.Multi-scale energy transfer in near-wall region of turbulentchannel flow.Chinese Journal of Theoretical and Applied Mechanics,2007,39(2):181-187(in Chinese))
13 Demongeot J,Bezy-Wendling J,Mattes J,et al.Multiscale modeling and imaging:the challenges of biocomplexity.Proceedings of the IEEE,2003,91(10):1723-1737
14 Savino GV,Formigli CM.Nonlinear electronic circuit with neuron like bursting and spiking dynamics.Biosystems,2009,97(1):9-14
15 Lu QS,Gu HG,Yang ZQ,et al.Dynamics of firing patterns,synchronization and resonances in neuronal electrical activities:Experiments and analysis.Acta Mech.Sin,2008,24(6):593-628
16 Sriram K,Gopinathan MS.Effects of delayed linear electrical perturbation of the Belousov-Zhabotinsky reaction:A case of complex mixed mode oscillations in a batch reactor.React.Kinet.Catal.Leu,2003,79(2):341-349
17 Wu HG,Bao BC,Liu Z,et al.Chaotic and period bursting phenomena in a memristive Wien-bridge oscillator.Nonlinear Dynamics,2016,83(1-2):893-903
18 Wang HX,Zheng YH,Lu QS.Stability and bifurcation analysis in the coupled HR neurons with delayed synaptic connection.Nonlinear Dynamics,2017,88(3):2091-2100
19陈振阳,韩修静,毕勤胜.一类二维非自治离散系统中的复杂簇发振荡结构.力学学报,2017,49(1):165-174(Chen Zhenyang,Han Xiujing,Bi Qinsheng.Complex bursting oscillation structures in a two-dimensional non-autonomous discrete system.Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):165-174(in Chinese))
20陈振阳,韩修静,毕勤胜.离散达芬映射中由边界激变所诱发的复杂的张弛振荡.力学学报,2017,49(6):1380-1389(Chen Zhenyang,Han Xiujing,Bi Qinsheng.Complex relaxation oscillation triggered by boundary crisis in the discrete Duffing map.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1380-1389(in Chinese))
21曲子芳,张正娣,彭淼等.双频激励下Filippov系统的非光滑簇发振荡机理.力学学报,2018,50(5):1145-1155(Qu Zifang,Zhang Zhengdi,Peng Miao,et al.Non-Smooth bursting oscillation mechanisms in a Filippov-type system with multiple periodic excitations.Chinese Journal of Theoretical and Applied Mechanics,2018,50(5):1145-1155(in Chinese))
22 Koper MTM.Bifurcations of mixed-mode oscillations in a threevariable autonomous van der Pol-Duffing model with a cross-shaped phase diagram.Physica D,1995,80(1):72-94
23毕勤胜,陈章耀,朱玉萍等.参数激励耦合系统的复杂动力学行为分析.力学学报,2003,35(3):367-372(Bi Qinsheng,Chen Zhangyao,Zhu Yuping,et al.Dynamical analysis of coupled oscillators with parametrical excitation.Chinese Journal of Theoretical and Applied Mechanics,2003,35(3):367-372(in Chinese))
24 Curtu R.Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network.Physica D,2010,239(9):504-514
25 Marino F,Ciszak M,Abdalah SF,et al.Mixed-mode oscillations via canard explosions in light-emitting diodes with optoelectronic feedback.Phys.Rev.E,2011,84(4 Pt 2):047201
26 Hou JY,Li XH,Zuo DW,et al.Bursting and delay behavior in the Belousov-Zhabotinsky reaction with external excitation.Eur.Phys.J.Plus,2017,132(6):283
27 Yu Y,Zhang C,Han XJ.Routes to bursting in active control system with multiple time delays.Nonlinear Dynamics,2017,88(3):2241-2254
28 Han XJ,Bi QS,Kurths J.Route to bursting via pulse-shaped explosion.Phys.Rev.E,2018,98(1):010201
29 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(1):012911
30 Rinzel J.Bursting oscillations in an excitable membrane model//Sleeman BD,Jarvis RJ.Ordinary and Partial Differential Equations.Berlin:Springer-Verlag,1985:304-316
31 Izhikevich EM.Neural excitability,spiking and bursting.Int.J.Bifurcation Chaos,2000,10(6):1171-1266
32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,2015,4(4):699-706(Chen Zhangyao,ChenYaguang,Bi Qinsheng.Bursting oscillation as well as the bifurcation mechanism induced by fast subsystem with multiple balances.Chinese Journal of Theoretical and Applied Mechanics,2015,4(4):699-706(in Chinese))
33 Stankevich N,Mosekilde E.Coexistence between silent and bursting states in a biophysical Hodgkin-Huxley-type of model.Chaos,2017,27(12):123101
34夏雨,毕勤胜,罗超等.双频1:2激励下修正蔡氏振子两尺度耦合行为.力学学报,2018,50(2):362-372(Xia Yu,Bi Qinsheng,Luo Chao,et al.Behaviors of modified Chua’s oscillator two time scales under two excitatoins with frequency ratio at 1:2.Chinese Journal of Theoretical and Applied Mechanics,2018,50(2):362-372(in Chinese))
35 Yu Y,Zhao M,Zhang ZD.Novel bursting patterns in a van der PolDuffing oscillator with slow varying external force.Mech.Syst.Signal Proc,2017,93:164-174