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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.
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Keywords:
- nonlinear time-delayed system /
- electronic circuits /
- Hopf bifurcation /
- chaos
[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
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[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
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