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一个简单延迟非线性系统的动力学行为及混沌同步

肖建新 陈菊芳 彭建华

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一个简单延迟非线性系统的动力学行为及混沌同步

肖建新, 陈菊芳, 彭建华

Dynamic behavior and chaos synchronization of a simple nonlinear time-delayed system

Xiao Jian-Xin, Chen Ju-Fang, Peng Jian-Hua
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  • 分析一个简单二阶延迟系统的Hopf分支和混沌特性, 包括分支点、分支方向和分支周期解的稳定性, 解析求出退延迟情况下, 这个系统的相轨线方程; 通过数值计算并绘制分岔图, 揭示系统存在由倍周期通向混沌的道路; 利用单路线性组合信号, 反馈控制实现系统的部分完全同步; 利用主动-被动与线性反馈的联合, 实现系统的完全同步; 设计和搭建系统的电子实验线路, 并从实验中观测到与理论分析或数值计算相一致的结果.
    Hopf bifurcation and chaotic properties of a simple second order time-delayed system, which includes bifrucation point, bifurcation direction and the stability of bifrucating periodic solutions, are analyzed. We obtain analytically the phase trajectory equations when the delay degenerates. Furthermore, through bifurcation diagram drawn by means of numerical simulation, the route from period-doubling bifurcation to chaos is reaveled; using single linearly combinating signals and the feedback control method, we achieve partial synchronization of the system. Combining the active-passive method with the linear feedback method, we have realized complete synchronization. In addition, we have designed and built an electronic experimental line, from which the same result as the theoretical analysis or numerical results are obtained.
    • 基金项目: 国家自然科学基金(批准号: 70571053)和吉林省科技发展计划(批准号: 201205001)资助的课题.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 70571053), and the Science and Technology Development Plan of Jilin Province of China (Grant No.201205001).
    [1]

    Mackey M C, Glass L 1977 Science 197 287

    [2]

    Tadenuma M U, Yamada R G, Xu H Y, Ripperger J A, Liu A C, Ueda H R 2011 Cell 144 268

    [3]

    Yao W, Yu P, Essex C 2001 Phys. Rev. E 63 021902

    [4]

    Masoller C 2001 Phys. Rev. Lett. 86 2782

    [5]

    Mo J Q, Wang H, Lin W T 2006 Acta Phys. Sin. 55 3230 (in Chinese) [莫嘉琪, 王辉, 林万涛 2006 物理学报 55 3230]

    [6]

    Hale J K 2002 Theory of functional differential equations (New York: Spring-Verlag)

    [7]

    Wei J J, Wang H B, Jiang W H 2012 Bifurcation theory and application of delay differential equations (Beijing: Science Press) (in Chinese) [魏俊杰, 王洪滨, 蒋卫华 2012 时滞微分方程的分支理论及应用 (北京: 科学出版社) ]

    [8]

    Larger L, Dudley J M 2010 Nature 465 41

    [9]

    Schanz M, Pelster A 2003 Phys. Rev. E 67 056205

    [10]

    Sprott J C 2007 Phys. Lett. A 366 397

    [11]

    Wang W, Xu Y, Lu S P 2010 Acta Phys. Sin. 60 030205 (in Chinese) [王雯, 徐燕, 鲁世平 2010 物理学报 60 030205]

    [12]

    Zhang X M, Chen J F, Peng J H 2011 International Journal of Bifurcation and Chaos 21 2547

    [13]

    Nguimdo R M, Colet P, Larger L, Pesquera L 2011 Phys. Rev. Lett. 107 034103

    [14]

    Yalcin M E, Ozoguz S 2007 Chaos 17 033112

    [15]

    Vasegh N, Khellat F 2009 Chaos Solit. Fract. 42 1054

    [16]

    Zhang X M 2004 M. S. Thesis (Changchun: Northest Normal University (in Chinese) [张晓明 2004年 硕士学位论文 (长春: 东北师范大学)]

    [17]

    Chen J F, Tian X J, San J D 2010 Acta Phys. Sin. 59 83 (in Chinese) [陈菊芳, 田小健, 单江东 2010 物理学报 59 83]

    [18]

    Peng J H, Ding E J, Yang W, Ding M 1996 Phys. Rev. Lett. 76 904

    [19]

    Cheng L, Zhang R Y, Peng J H 2003 Acta Phys.Sin. 52 536 (in Chinese) [程丽, 张入元, 彭建华 2003 物理学报 52 536]

    [20]

    Kocarev L, Parlitz U 1995 Phys. Rev. Lett. 74 5208

    [21]

    Chen J F, Zhang R Y, Peng J H 2003 Acta Phys. Sin. 52 1589 (in Chinese) [陈菊芳, 张入元, 彭建华 2003 物理学报 52 1589]

    [22]

    Ding M, Ott E 1994 Phys. Rev. E 49 R945

    [23]

    Corless R M, Gonnet G H, Hare D E G, Jeffrey D J, Knuth D E 1996 Adv. Comput. Math. 5 329

  • [1]

    Mackey M C, Glass L 1977 Science 197 287

    [2]

    Tadenuma M U, Yamada R G, Xu H Y, Ripperger J A, Liu A C, Ueda H R 2011 Cell 144 268

    [3]

    Yao W, Yu P, Essex C 2001 Phys. Rev. E 63 021902

    [4]

    Masoller C 2001 Phys. Rev. Lett. 86 2782

    [5]

    Mo J Q, Wang H, Lin W T 2006 Acta Phys. Sin. 55 3230 (in Chinese) [莫嘉琪, 王辉, 林万涛 2006 物理学报 55 3230]

    [6]

    Hale J K 2002 Theory of functional differential equations (New York: Spring-Verlag)

    [7]

    Wei J J, Wang H B, Jiang W H 2012 Bifurcation theory and application of delay differential equations (Beijing: Science Press) (in Chinese) [魏俊杰, 王洪滨, 蒋卫华 2012 时滞微分方程的分支理论及应用 (北京: 科学出版社) ]

    [8]

    Larger L, Dudley J M 2010 Nature 465 41

    [9]

    Schanz M, Pelster A 2003 Phys. Rev. E 67 056205

    [10]

    Sprott J C 2007 Phys. Lett. A 366 397

    [11]

    Wang W, Xu Y, Lu S P 2010 Acta Phys. Sin. 60 030205 (in Chinese) [王雯, 徐燕, 鲁世平 2010 物理学报 60 030205]

    [12]

    Zhang X M, Chen J F, Peng J H 2011 International Journal of Bifurcation and Chaos 21 2547

    [13]

    Nguimdo R M, Colet P, Larger L, Pesquera L 2011 Phys. Rev. Lett. 107 034103

    [14]

    Yalcin M E, Ozoguz S 2007 Chaos 17 033112

    [15]

    Vasegh N, Khellat F 2009 Chaos Solit. Fract. 42 1054

    [16]

    Zhang X M 2004 M. S. Thesis (Changchun: Northest Normal University (in Chinese) [张晓明 2004年 硕士学位论文 (长春: 东北师范大学)]

    [17]

    Chen J F, Tian X J, San J D 2010 Acta Phys. Sin. 59 83 (in Chinese) [陈菊芳, 田小健, 单江东 2010 物理学报 59 83]

    [18]

    Peng J H, Ding E J, Yang W, Ding M 1996 Phys. Rev. Lett. 76 904

    [19]

    Cheng L, Zhang R Y, Peng J H 2003 Acta Phys.Sin. 52 536 (in Chinese) [程丽, 张入元, 彭建华 2003 物理学报 52 536]

    [20]

    Kocarev L, Parlitz U 1995 Phys. Rev. Lett. 74 5208

    [21]

    Chen J F, Zhang R Y, Peng J H 2003 Acta Phys. Sin. 52 1589 (in Chinese) [陈菊芳, 张入元, 彭建华 2003 物理学报 52 1589]

    [22]

    Ding M, Ott E 1994 Phys. Rev. E 49 R945

    [23]

    Corless R M, Gonnet G H, Hare D E G, Jeffrey D J, Knuth D E 1996 Adv. Comput. Math. 5 329

计量
  • 文章访问数:  5253
  • PDF下载量:  663
  • 被引次数: 0
出版历程
  • 收稿日期:  2013-04-02
  • 修回日期:  2013-05-17
  • 刊出日期:  2013-09-05

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