时滞耦合van der Pol-Duffing振子环的动力学分析
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摘要
研究含时滞的大规模van der Pol-Duffing耦合振子系统的非线性动力学.通过讨论特征方程根分布情况确定系统的稳定性,并在耦合时滞和强度平面上给出振幅死亡区域.结合数值算例,揭示同步和异步周期振荡、概周期运动以及混沌吸引子等现象.基于非线性振子电路和时滞电路,构建电路实验平台,有效验证理论和数值结果.研究结果表明,时滞可以显著影响系统动力学特性,如诱发振幅死亡、稳定性切换以及复杂振荡等.
The dynamic behaviors of a delay-coupled ring of van der Pol-Duffing oscillators were studied. The stability and bifurcation of the system were determined by solving the associated characteristic equation. The parametrical regions of amplitude death were shown in the plane of time delay and coupling strength. Case studies were carried out by numerical simulations, which were validated by circuit experiments. It was shown that time delay can give rise to abundant and interesting behaviors, such as amplitude death, different periodic oscillations, quasi-periodic responses, and even chaotic attractors.
The dynamic behaviors of a delay-coupled ring of van der Pol-Duffing oscillators were studied. The stability and bifurcation of the system were determined by solving the associated characteristic equation. The parametrical regions of amplitude death were shown in the plane of time delay and coupling strength. Case studies were carried out by numerical simulations, which were validated by circuit experiments. It was shown that time delay can give rise to abundant and interesting behaviors, such as amplitude death, different periodic oscillations, quasi-periodic responses, and even chaotic attractors.
引文
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13 Jiang H,Zhang T H,Song Y L.Delay-induced double Hopf bifurcations in a system of two delay-coupled van der Pol-Duffing oscillators.International Journal of Bifurcation and Chaos,2015,25(4):1550058
14 Liu X,Zhang T H.Bogdanov-takens and triple zero bifurcations of coupled van der Pol-Duffing oscillators with multiple delays.International Journal of Bifurcation and Chaos,2017,27(9):1750133
15 Nana B,Woafo P.Synchronized states in a ring of four mutually coupled oscillators and experimental application to secure communications.Communications in Nonlinear Science and Numerical Simulation,2011,16(4):1725～1733
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17 Ablay G.Novel chaotic delay systems and electronic circuit solutions.Nonlinear Dynamics,2015,81(4):1795～1804
2 Wang Q Y,Zheng Y H,Ma J.Cooperative dynamics in neuronal networks.Chaos,Solitons & Fractals,2013,56:19～27
3 Song Y L,Xu J,Zhang T H.Bifurcation,amplitude death and oscillation patterns in a system of three coupled van der Pol oscillators with diffusively delayed velocity coupling.Chaos,2011,21(2):023111
4 Gjurchinovski A,Zakharova A,Sch?ll E.Amplitude death in oscillator networks with variable-delay coupling.Physical Review E,2014,89(3):032915
5 Hu H Y,Wang Z H.Dynamics of controlled mechanical systems with delayed feedback.Springer:Springer Science & Business Media,2013
6 王在华,胡海岩.时滞动力系统的稳定性与分岔:从理论走向应用.力学进展,2013,43(1):3～20 (Wang Z H,Hu H Y.Stability and bifurcation of delayed dynamic systems:from theory to application.Advances in Mechanics,2013,43(1):3～20 (in Chinese))
7 张舒,徐鉴.时滞耦合系统非线性动力学的研究进展.力学学报,2017,49(3):565～587 (Zhang S,Xu J.Review on nonlinear dynamics in systems with coupling delays.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):565～587 (in Chinese))
8 薛焕斌,张继业.时滞切换不确定神经网络系统的指数稳定性.动力学与控制学报,2018,16(1):65～71 (Xue H B,Zhang J Y.Exponential stability of time-delay switched uncertain neural networks systems.Journal of Dynamics and Control,2018,16(1):65～71 (in Chinese))
9 Brezetskyi S,Dudkowski D,Kapitaniak T.Rare and hidden attractors in Van der Pol-Duffing oscillators.European Physical Journal Special Topics,2015,224(8):1459～1467
10 Xu J,Chung K W.Effects of time delayed position feedback on a van der Pol-Duffing oscillator.Physica D,2003,180(1-2):17～39
11 Maccari A.Vibration amplitude control for a van der Pol-Duffing oscillator with time delay.Journal of Sound & Vibration,2008,317(1-2):20～29
12 Zang H,Zhang T H,Zhang Y D.Stability and bifurcation analysis of delay coupled Van der Pol-Duffing oscillators.Nonlinear Dynamics,2014,75(1-2):35～47
13 Jiang H,Zhang T H,Song Y L.Delay-induced double Hopf bifurcations in a system of two delay-coupled van der Pol-Duffing oscillators.International Journal of Bifurcation and Chaos,2015,25(4):1550058
14 Liu X,Zhang T H.Bogdanov-takens and triple zero bifurcations of coupled van der Pol-Duffing oscillators with multiple delays.International Journal of Bifurcation and Chaos,2017,27(9):1750133
15 Nana B,Woafo P.Synchronized states in a ring of four mutually coupled oscillators and experimental application to secure communications.Communications in Nonlinear Science and Numerical Simulation,2011,16(4):1725～1733
16 Orosz G,Wilson R E,Stépán G.Traffic jams:dynamics and control.Philosophical Transactions Mathematical Physical & Engineering Sciences,2010,368(1928):4455～4479
17 Ablay G.Novel chaotic delay systems and electronic circuit solutions.Nonlinear Dynamics,2015,81(4):1795～1804