一类混沌系统中的簇发振荡及其延迟叉形分岔行为
|
摘要
由于多时间尺度问题在实际工程系统中广泛存在,关于其复杂动力学行为及其产生机制的研究已成为当前国内外的热点课题之一.簇发振荡是多时间尺度系统复杂动力学行为的典型代表,而分岔延迟又是簇发振荡中的常见现象.本文为探讨非线性系统中分岔延迟所引发的簇发振荡的分岔机制,在一个三维混沌系统中引入参数激励,当激励频率远小于系统的固有频率时,系统产生了两时间尺度簇发振荡.将整个激励项看做慢变参数,激励系统转化为广义自治系统也即快子系统,分析快子系统平衡点的稳定性以及分岔条件,并运用快慢分析法和转换相图揭示了簇发振荡的动力学机理.文中考察了4组参数条件下系统的动力学行为,研究发现当慢变激励项周期性地通过分岔点时,系统产生了明显的超临界叉形分岔延迟行为,随着参数激励振幅的增大,分岔延迟的时间也逐渐延长,当这种延迟的动态行为终止于不同的参数区域时,导致系统轨线围绕不同稳定吸引子(平衡点,极限环)运动,从而得到了不同的簇发振荡行为.
Due to wide existence of multiple-time-scale problems in practical engineering, the complicated dynamic behaviors and their generation mechanism have become one of the hot topics at home and abroad. The systems with multiple time scales can often exhibit bursting oscillations with the bifurcation delay phenomenon. In order to investigate the bifurcation mechanism of bursting oscillations caused by bifurcation delay in a nonlinear system, a parametric excitation is introduced in a novel three-dimensional chaotic system. When the exciting frequency is far less than the natural frequency, the coupling of two time scales involves the vector field, which leads to the bursting oscillations. By considering the whole exciting term as a slow-varying parameter, the original system can be considered as a generalized autonomous system, which can be regarded as the fast subsystem. Upon the analysis of equilibrium points and bifurcation conditions of the fast subsystem, combining with the transformed phase portraits, the bifurcation mechanisms of bursting oscillations is presented. Four typical cases with different parameter conditions are discussed to reveal the evolution of the bursting oscillations. It is found that when the slow-varying exciting term passes across the bifurcation points, the delayed behaviors of super-critical pitchfork bifurcation can be observed. With the increase of the exciting amplitude, the occurring needed for the bifurcation delay is increased gradually. When the delayed behaviors end in different parameter regions, different types of bursting oscillations which may surround different attractors such as equilibrium points and limit cycles appear.
Due to wide existence of multiple-time-scale problems in practical engineering, the complicated dynamic behaviors and their generation mechanism have become one of the hot topics at home and abroad. The systems with multiple time scales can often exhibit bursting oscillations with the bifurcation delay phenomenon. In order to investigate the bifurcation mechanism of bursting oscillations caused by bifurcation delay in a nonlinear system, a parametric excitation is introduced in a novel three-dimensional chaotic system. When the exciting frequency is far less than the natural frequency, the coupling of two time scales involves the vector field, which leads to the bursting oscillations. By considering the whole exciting term as a slow-varying parameter, the original system can be considered as a generalized autonomous system, which can be regarded as the fast subsystem. Upon the analysis of equilibrium points and bifurcation conditions of the fast subsystem, combining with the transformed phase portraits, the bifurcation mechanisms of bursting oscillations is presented. Four typical cases with different parameter conditions are discussed to reveal the evolution of the bursting oscillations. It is found that when the slow-varying exciting term passes across the bifurcation points, the delayed behaviors of super-critical pitchfork bifurcation can be observed. With the increase of the exciting amplitude, the occurring needed for the bifurcation delay is increased gradually. When the delayed behaviors end in different parameter regions, different types of bursting oscillations which may surround different attractors such as equilibrium points and limit cycles appear.
引文
1 Li XH,Hou JY,Chen JF.An analytical method for Mathieu oscillator based on method of variation of parameter.Communications in Nonlinear Science&Numerical Simulation,2016,37:326-353
2 Dai HH,Yue XK,Liu C.A multiple scale time domain collocation method for solving nonlinear dynamical system.International Journal of Nonlinear Mechanics,2014,67:342-351
3张洪武,张盛,毕金英.周期性结构热动力时间-空间多尺度分析.力学学报,2006,38(2):226-235(Zhang Hongwu,Zhang Sheng,Bi Jinying.Thermodynamic analysis of multiphase periodic structures based on a spatial and temporal multiple scale method.Chinese Journal of Theoretical and Applied Mechanics,2006,38(2):226-235(in Chinese))
4 Br?ns M,Kaasen R.Canards and mixed-mode oscillations in a forest pest model.Theoretical Population Biology,2010,77(4):238-242
5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 Guaguang.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))
6 Liu X,Huo B.Nonlinear vibration and multimodal interaction analysis of transmission line with thin ice accretions.International Journal of Applied Mechanics,2015,7(1):1540007
7 Li XH,Bi QS.Single-Hopf bursting in periodic perturbed BelousovZhab-otinsky reaction with two time scales.Chin.phys.lett,2013,30(1):10503-10506
8李向红,毕勤胜.铂族金属氧化过程中的簇发振荡及其诱发机理.物理学报,2012,61(2):88-96(Li Xianghong,Bi Qinsheng.Bursting oscillations and the bifurcation mechanism in oxidation on platinum group metals.Acta Phys Sin,2012,61(2):88-96(in Chinese))
9 Courbage M,Nekorkin VI,Vdovin LV.Chaotic oscillations in a map-based model of neural activity.Chaos,2007,17(4):043109
10 Izhikevich EM.Resonance and selective communication via bursts in neurons having subthreshold oscillations.Bio Systems,2002,67(1-3):95-102
11 Ferrari FAS,Viana RL,Lopes SR,et al.Phase synchronization of coupled bursting neurons and the generalized Kuramoto model.Neural Networks,2015,66:107-118
12 Bi QS.The mechanism of bursting phenomena in BZ chemical reaction with multiple time scales.Science in China,2012,10:2820-2830
13陈章耀,张晓芳,毕勤胜.周期激励下Hartley模型的簇发及分岔机.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))
14 Farazmand M,Sapsis T.Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems.Physical Review E,2016,94(3):032212
15 Wu H,Bao BC,Liu Z,et al.Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator.Nonlinear Dynamics,2016,83(1-2):893-903
16 Fujimoto K,Kaneko K.How fast elements can affect slow dynamics.Physica D,2003,180(1):1-16
17 Egghea L,Rousseaub R.Lorenz theory of symmetric relative concentration and similarity,incorporating variable array length.Mathematical&Computer Modelling,2006,44(7):628-639
18 Stork W.Subharmonic response and synchronisation for the forced van der Pol equation via Cesari’s method.Journal of Mathematical Analysis&Applications,1984,102(1):134-148
19 Izhikevich EM.Neural excitability,spiking and bursting.International Journal of Bifurcation and Chaos,2000,10(6):1171-1266
20 Bi QS,Li SL,Kurths J,et al.The mechanism of bursting oscillations with different codimensional bifurcations and nonlinear structures.Nonlinear Dynamics,2016,85(2):1-13
21 Han XJ,Bi QS,Ji P,et al.Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies.Physical Review E,2015,92(1):012911
22 Yu Y,Tang HJ,Han XJ,et al.Bursting mechanism in a time-delayed oscillator with slowly varying external forcing.Communications in Nonlinear Science&Numerical Simulation,2014,19(4):1175-1184
23张晓芳,陈小可,毕勤胜.快慢耦合振子的张驰簇发及其非光滑分岔机制.力学学报,2012,44(3):576-583(Zhang Xiaofang,Chen Xiaoke,Bi Qinsheng.Relaxation bursting of a fast-slow coupled oscillation as well as the mechanism of non-smooth bifurcation.Chinese Journal of Theoretical and Applied Mechanics,2012,44(3):576-583(in Chinese))
24 Freire JG,Gallas JAC.Stern-Brocot trees in cascades of mixedmode oscillations and canards in the extended Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of excitable systems.Phys Lett A,2011,375(7):1097-1103
25 Chumakov GA,Chumakova NA.Relaxation oscillations in a kinetic model of catalytic hydrogen oxidation involving a chase on canards.Chemical Engineering Journal,2003,91(2):151-158
26韩修静,江波,毕勤胜.快慢型超混沌Lorenz系统分析.物理学报,2009,58(9):6006-6015(Han Xiujing,Jiang Bo,Bi Qinsheng.Analysis of the fast-slow hyperchaotic Lorenz system.Acta Phys Sin,2009,58(9):6006-6015(in Chinese))
27 Hou JY,Li XH,Zuo DW,et al.Bursting and delay behavior in the Belousov-Zhabotinsky reaction with external excitation.European Physical Journal Plus,2017,132(6):283
28 Baer SM,Erneux T,Rinzel J.The slow passage through a Hopf bifurcation:Delay,memory effects,and resonance.Siam Journal on Applied Mathematics,1989,49(1):55-71
29 Hao LJ,Yang ZQ,Lei JZ.Bifurcation analysis of a delay differential equation model associated with the induction of long-term memory.Chaos Solitons&Fractals,2015,81:162-171
30 Mahmoud GM,Arafa AA,Mahmoud EE.Bifurcations and chaos of time delay Lorenz system with dimension 2n+1.European Physical Journal Plus,2017,132(1):461
31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))
32 Maree GJM.Slow passage through a pitchfork bifurcation.SIAMJournal on Applied Mathematics,1996,56(3):889-918
33 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.Communications in Nonlinear Science&Numerical Simulation,2016,37:212-221
34 Yu Y,Han XJ,Zhang C,et al.Mixed-mode oscillations in a nonlinear time delay oscillator with time varying parameters.Communications in Nonlinear Science&Numerical Simulation,2017,47:23-34
35陈章耀,王亚茗,张春等.双状态切换下BVP振子的复杂行为分析.力学学报,2016,48(4):953-962(Chen Zhangyao,Wang Yaming,Zhang Chun,et al.Complicated behaviors as well as the mechanism in BVP oscillator with switches related to two states.Chinese Journal of Theoretical and Applied Mechanics,2016,48(4):953-962(in Chinese))
36 Abooee A,Yaghini-Bonabi HA,Jahed-Motlagh MR.Analysis and circuitry realization of a novel three-dimensional chaotic system.Communications in Nonlinear Science&Numerical Simulation,2013,18(5):1235-1245
2 Dai HH,Yue XK,Liu C.A multiple scale time domain collocation method for solving nonlinear dynamical system.International Journal of Nonlinear Mechanics,2014,67:342-351
3张洪武,张盛,毕金英.周期性结构热动力时间-空间多尺度分析.力学学报,2006,38(2):226-235(Zhang Hongwu,Zhang Sheng,Bi Jinying.Thermodynamic analysis of multiphase periodic structures based on a spatial and temporal multiple scale method.Chinese Journal of Theoretical and Applied Mechanics,2006,38(2):226-235(in Chinese))
4 Br?ns M,Kaasen R.Canards and mixed-mode oscillations in a forest pest model.Theoretical Population Biology,2010,77(4):238-242
5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 Guaguang.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))
6 Liu X,Huo B.Nonlinear vibration and multimodal interaction analysis of transmission line with thin ice accretions.International Journal of Applied Mechanics,2015,7(1):1540007
7 Li XH,Bi QS.Single-Hopf bursting in periodic perturbed BelousovZhab-otinsky reaction with two time scales.Chin.phys.lett,2013,30(1):10503-10506
8李向红,毕勤胜.铂族金属氧化过程中的簇发振荡及其诱发机理.物理学报,2012,61(2):88-96(Li Xianghong,Bi Qinsheng.Bursting oscillations and the bifurcation mechanism in oxidation on platinum group metals.Acta Phys Sin,2012,61(2):88-96(in Chinese))
9 Courbage M,Nekorkin VI,Vdovin LV.Chaotic oscillations in a map-based model of neural activity.Chaos,2007,17(4):043109
10 Izhikevich EM.Resonance and selective communication via bursts in neurons having subthreshold oscillations.Bio Systems,2002,67(1-3):95-102
11 Ferrari FAS,Viana RL,Lopes SR,et al.Phase synchronization of coupled bursting neurons and the generalized Kuramoto model.Neural Networks,2015,66:107-118
12 Bi QS.The mechanism of bursting phenomena in BZ chemical reaction with multiple time scales.Science in China,2012,10:2820-2830
13陈章耀,张晓芳,毕勤胜.周期激励下Hartley模型的簇发及分岔机.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))
14 Farazmand M,Sapsis T.Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems.Physical Review E,2016,94(3):032212
15 Wu H,Bao BC,Liu Z,et al.Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator.Nonlinear Dynamics,2016,83(1-2):893-903
16 Fujimoto K,Kaneko K.How fast elements can affect slow dynamics.Physica D,2003,180(1):1-16
17 Egghea L,Rousseaub R.Lorenz theory of symmetric relative concentration and similarity,incorporating variable array length.Mathematical&Computer Modelling,2006,44(7):628-639
18 Stork W.Subharmonic response and synchronisation for the forced van der Pol equation via Cesari’s method.Journal of Mathematical Analysis&Applications,1984,102(1):134-148
19 Izhikevich EM.Neural excitability,spiking and bursting.International Journal of Bifurcation and Chaos,2000,10(6):1171-1266
20 Bi QS,Li SL,Kurths J,et al.The mechanism of bursting oscillations with different codimensional bifurcations and nonlinear structures.Nonlinear Dynamics,2016,85(2):1-13
21 Han XJ,Bi QS,Ji P,et al.Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies.Physical Review E,2015,92(1):012911
22 Yu Y,Tang HJ,Han XJ,et al.Bursting mechanism in a time-delayed oscillator with slowly varying external forcing.Communications in Nonlinear Science&Numerical Simulation,2014,19(4):1175-1184
23张晓芳,陈小可,毕勤胜.快慢耦合振子的张驰簇发及其非光滑分岔机制.力学学报,2012,44(3):576-583(Zhang Xiaofang,Chen Xiaoke,Bi Qinsheng.Relaxation bursting of a fast-slow coupled oscillation as well as the mechanism of non-smooth bifurcation.Chinese Journal of Theoretical and Applied Mechanics,2012,44(3):576-583(in Chinese))
24 Freire JG,Gallas JAC.Stern-Brocot trees in cascades of mixedmode oscillations and canards in the extended Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of excitable systems.Phys Lett A,2011,375(7):1097-1103
25 Chumakov GA,Chumakova NA.Relaxation oscillations in a kinetic model of catalytic hydrogen oxidation involving a chase on canards.Chemical Engineering Journal,2003,91(2):151-158
26韩修静,江波,毕勤胜.快慢型超混沌Lorenz系统分析.物理学报,2009,58(9):6006-6015(Han Xiujing,Jiang Bo,Bi Qinsheng.Analysis of the fast-slow hyperchaotic Lorenz system.Acta Phys Sin,2009,58(9):6006-6015(in Chinese))
27 Hou JY,Li XH,Zuo DW,et al.Bursting and delay behavior in the Belousov-Zhabotinsky reaction with external excitation.European Physical Journal Plus,2017,132(6):283
28 Baer SM,Erneux T,Rinzel J.The slow passage through a Hopf bifurcation:Delay,memory effects,and resonance.Siam Journal on Applied Mathematics,1989,49(1):55-71
29 Hao LJ,Yang ZQ,Lei JZ.Bifurcation analysis of a delay differential equation model associated with the induction of long-term memory.Chaos Solitons&Fractals,2015,81:162-171
30 Mahmoud GM,Arafa AA,Mahmoud EE.Bifurcations and chaos of time delay Lorenz system with dimension 2n+1.European Physical Journal Plus,2017,132(1):461
31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))
32 Maree GJM.Slow passage through a pitchfork bifurcation.SIAMJournal on Applied Mathematics,1996,56(3):889-918
33 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.Communications in Nonlinear Science&Numerical Simulation,2016,37:212-221
34 Yu Y,Han XJ,Zhang C,et al.Mixed-mode oscillations in a nonlinear time delay oscillator with time varying parameters.Communications in Nonlinear Science&Numerical Simulation,2017,47:23-34
35陈章耀,王亚茗,张春等.双状态切换下BVP振子的复杂行为分析.力学学报,2016,48(4):953-962(Chen Zhangyao,Wang Yaming,Zhang Chun,et al.Complicated behaviors as well as the mechanism in BVP oscillator with switches related to two states.Chinese Journal of Theoretical and Applied Mechanics,2016,48(4):953-962(in Chinese))
36 Abooee A,Yaghini-Bonabi HA,Jahed-Motlagh MR.Analysis and circuitry realization of a novel three-dimensional chaotic system.Communications in Nonlinear Science&Numerical Simulation,2013,18(5):1235-1245