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研究了耦合分数阶振子的同步、反同步和振幅死亡等问题. 基于P-R振子在特定参数下的双稳态特性, 利用最大条件Lyapunov指数、最大Lyapunov指数和分岔图等数值方法分析发现, 通过选取初始条件和耦合强度, 可以控制耦合振子呈现混沌同步、混沌反同步、全部振幅死亡同步、全部振幅死亡反同步和部 分振幅死亡等丰富的动力学现象. 基于蒙特卡罗方法的原理, 在初始条件相空间中随机选取耦合振子的初始位置, 计算不同耦合强度下耦合振子的全部振幅死亡态、部分振幅死亡态和非振幅死亡态的比例, 从统计学角度表征了耦合分数阶双稳态振子的动力学特征. 几种有代表性的双稳态振子的吸引域进一步证明了统计方法的计算结果.The dynamic behaviors of coupled fractional order bistable oscillators are investigated extensively and various phenomena such as synchronization, anti-synchronization, and amplitude death, etc. are explored. Based on the bistable characteristics of P-R oscillator with specific parameters, effects of initial conditions and coupling strength on the dynamic behaviors of the coupled fractional order bistable oscillators are first investigated by analyzing the maximum condition of Lyapunov exponent, the maximum Lyapunov exponent and the bifurcation diagram, etc. Further investigation reveals that the coupled fractional order bistable oscillators can be controlled to form chaotic synchronization, chaotic anti-synchronization, synchronous amplitude death, anti-synchronous amplitude death, partial amplitude death, and so on by changing the initial conditions and the coupling strength. Then, based on the principle of Monte Carlo method, by randomly choosing the initial conditions from the phase space, we calculate the percentage of various states when changing the coupling strength, so the dynamic characteristics of coupled fractional-order bistable oscillators can be represented by using the perspective of statistics. Some representative attractive basins are plotted, which are well coincident with numerical simulations.
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Keywords:
- amplitude death /
- attractive basin /
- bistable state
[1] Matthews P C, Strogatz S H 1990 Phys. Rev. Lett. 65 1701
[2] Ramana Reddy D V, Sen A, Johnston G L 1998 Phys. Rev. Lett. 80 5109
[3] Konishi K 2004 Phys. Rev. E 70 066201
[4] Konishi K, Senda K, Kokame H 2008 Phys. Rev. E 78 056216
[5] Prasad A, Dhamala M, Adhikari B M, Ramaswamy R 2010 Phys. Rev. E 81 027201
[6] Zhu Y, Qian X L, Yang J Z 2008 Europhys. Lett. 82 40001
[7] Shao S Y, Min F H, Ma M L, Wang E R 2013 Acta Phys. Sin. 62 130504 (in Chinese) [邵书义, 闵富红, 马美玲, 王恩荣 2013 物理学报 62 130504]
[8] Jia H Y, Chen Z Q, Xue W 2013 Acta Phys. Sin. 62 140503 (in Chinese) [贾红艳, 陈增强, 薛薇 2013 物理学报 62 140503]
[9] Shao S Q, Gao X, Liu X W 2007 Acta Phys. Sin. 56 6815 (in chinese) [邵仕泉, 高心, 刘兴文 2007 物理学报 56 6815]
[10] Deng W H, Li C P 2005 Physica A 353 61
[11] Chen X R, Liu C X, Wang F Q, Li Y X 2008 Acta Phys. Sin. 57 1416 (in Chinese) [陈向荣, 刘崇新, 王发强, 李永勋 2008 物理学报 57 1416]
[12] Podlubny I 1999 Fractional Differential aligns 198 (San Diego: Academic Press) p78
[13] Deng W H, Li C P, L J H 2007 Nonlinear Dyn. 48 409
[14] Zhang R X, Yang S P 2011 Chin. Phys. B 20 110506
[15] Li C P, Peng G J 2004 Chaos, Solitons and Fractals 22 443
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[1] Matthews P C, Strogatz S H 1990 Phys. Rev. Lett. 65 1701
[2] Ramana Reddy D V, Sen A, Johnston G L 1998 Phys. Rev. Lett. 80 5109
[3] Konishi K 2004 Phys. Rev. E 70 066201
[4] Konishi K, Senda K, Kokame H 2008 Phys. Rev. E 78 056216
[5] Prasad A, Dhamala M, Adhikari B M, Ramaswamy R 2010 Phys. Rev. E 81 027201
[6] Zhu Y, Qian X L, Yang J Z 2008 Europhys. Lett. 82 40001
[7] Shao S Y, Min F H, Ma M L, Wang E R 2013 Acta Phys. Sin. 62 130504 (in Chinese) [邵书义, 闵富红, 马美玲, 王恩荣 2013 物理学报 62 130504]
[8] Jia H Y, Chen Z Q, Xue W 2013 Acta Phys. Sin. 62 140503 (in Chinese) [贾红艳, 陈增强, 薛薇 2013 物理学报 62 140503]
[9] Shao S Q, Gao X, Liu X W 2007 Acta Phys. Sin. 56 6815 (in chinese) [邵仕泉, 高心, 刘兴文 2007 物理学报 56 6815]
[10] Deng W H, Li C P 2005 Physica A 353 61
[11] Chen X R, Liu C X, Wang F Q, Li Y X 2008 Acta Phys. Sin. 57 1416 (in Chinese) [陈向荣, 刘崇新, 王发强, 李永勋 2008 物理学报 57 1416]
[12] Podlubny I 1999 Fractional Differential aligns 198 (San Diego: Academic Press) p78
[13] Deng W H, Li C P, L J H 2007 Nonlinear Dyn. 48 409
[14] Zhang R X, Yang S P 2011 Chin. Phys. B 20 110506
[15] Li C P, Peng G J 2004 Chaos, Solitons and Fractals 22 443
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