具非均匀剪切项的耦合振子系统的同步分析
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摘要
讨论了具非均匀剪切项的全局耦合相位振子系统由非同步状态向同步状态转变的动力学问题,其中剪切项的强度服从洛伦兹分布。利用改进的Kuramoto模型,通过运用OA约化方法,得到了相位振子系统的约化方程,进而利用微分方程的稳定性理论,得到了系统标准差阈值,这里得到标准差阈值方法与现有文献中通过构造拟设的方法所求得标准差阈值方法更为一般,此外还通过作出约化方程的波形图、相图以及序参量随时间变化的图像来进一步验证所得结果。当剪切强度标准差超过该阈值时,系统在任意耦合强度下不可能同步,当剪切强度标准差小于该阈值时,系统在耦合强度超过某一定值时会产生同步。这与在无剪切项作用下,系统可以通过调整耦合强度最终达到同步状态的结果不同,说明了剪切项对耦合相位振子系统具有重要的影响。
The dynamics of the global coupled phase oscillator system with an inhomogeneous shear term is studied. By using the improved Kuramoto model and OA reduction method, the reduction equation of the discussed phase oscillator system is obtained, and then the standard deviation threshold of the system is obtained by using the stability theory of the differential equations. When the standard deviation of shear strength exceeds the threshold value, the system cannot be synchronized under any coupling strength, while the standard deviation of shear strength is less than the threshold value,the system will produce synchronization when the coupling strength exceeds a certain value. This is different from the result that the system can reach the final synchronous state by adjusting the coupling strength under the action of no shear term, which indicates that the shear term has an important influence on the coupled phase vibration system.
The dynamics of the global coupled phase oscillator system with an inhomogeneous shear term is studied. By using the improved Kuramoto model and OA reduction method, the reduction equation of the discussed phase oscillator system is obtained, and then the standard deviation threshold of the system is obtained by using the stability theory of the differential equations. When the standard deviation of shear strength exceeds the threshold value, the system cannot be synchronized under any coupling strength, while the standard deviation of shear strength is less than the threshold value,the system will produce synchronization when the coupling strength exceeds a certain value. This is different from the result that the system can reach the final synchronous state by adjusting the coupling strength under the action of no shear term, which indicates that the shear term has an important influence on the coupled phase vibration system.
引文
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[4] JOHN Buck E. Synchronous fireflies[J]. Scientific American,1976,234(5):74-85.
[5] WIESENFELD K,COLET P,STROGATZ S H. Synchronization transitions in a disordered Josephson series array[J]. Physical Review Letters,1996,76(3):404.
[6] STROGATZ S. Review of Sync:The emerging science of spontaneous order[J]. Notices of the American Mathematical Society,2003(3).
[7] WIENER N. Nonlinear problems in random theory[J]. Students Quarterly Journal,1959,30(118):77.
[8] WINFREE A T. Biological rhythms and the behavior of populations of coupled oscillators[J]. Journal of Theoretical Biology,1967,16(1):15-42.
[9] KURAMOTO Y. Chemical Oscillations,Waves,and Turbulence[M]. Berlin,Heidelberg:Springer Berlin Heidelberg,1984:355-388.
[10] STROGATZ S H. From Kuramoto to Crawford:exploring the onset of synchronization in populations of coupled oscillators[J]. Physica D:Nonlinear Phenomena,2000,143(1/2/3/4):1-20.
[11] KURAMOTO F,NISHIKAWA I. Onset of collective rhythms in large populations of coupled oscillators[C]//Kuramoto F,Nishikawa I. Springer Series in Synergetics. Berlin,Heidelberg:Springer Berlin Heidelberg,1989:300-306.
[12] TANAKA H A,LICHTENBERG A J,OISHI S. First order phase transition resulting from finite inertia in coupled oscillator systems[J]. Physical Review Letters,1997,78(11):2104-2107.
[13] HONG H,CHOI M Y,YI J,et al. Inertia effects on periodic synchronization in a system of coupled oscillators[J]. Physical Review E,1999,59(1):353-363.
[14]黄霞,徐灿,孙玉庭,等.耦合振子系统的多稳态同步分析[J].物理学报,2015,64(17):170504-1-170504-11.
[15] VLASOV V,MACAU E E N,PIKOVSKY A. Synchronization of oscillators in a Kuramoto-type model with generic coupling[J]. Chaos:an Interdisciplinary Journal of Nonlinear Science,2014,24(2):023120.
[16] KURAMOTO Y. International Symposium on Mathematical Problems in Theoretical Physics,Lecture Notes in Physics[C]//3rd IEEE International Symposium on Biomedical Imaging:Nano to Macro,2006.
[17] CROSS M C,ZUMDIECK A,LIFSHITZ R,et al. Synchronization by nonlinear frequency pulling.[J]. Physical Review Letters,2004,93(22):224101.
[18] MONTBRIóE,PAZóD. Shear diversity prevents collective synchronization[J]. Physical Review Letters,2011,106(25):254101.
[19] CROSS M C,HOHENBERG P C. Pattern formation outside of equilibrium[J]. Review of Modern Physics,1993,65(3):851-1112.
[20] STROGATZ S H,MIROLLO R E. Stability of incoherence in a population of coupled oscillators[J]. Journal of Statistical Physics,1991,63(3/4):613-635.
[21] OTT E,ANTONSEN T M. Long time evolution of phase oscillator systems[J]. Chaos:an Interdisciplinary Journal of Nonlinear Science,2009,19(2):023117.
[22] OTT E,ANTONSEN T M. Low dimensional behavior of large systems of globally coupled oscillators[J]. Chaos:an Interdisciplinary Journal of Nonlinear Science,2008,18(3):037113.
[23]陆启韶,彭临平,杨卓琴.常微分方程与动力系统[M].北京:北京航空航天大学出版社,2010.