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## 留言板

Non-smooth bifurcation in nonlinear electrical circuits with multiple switching boundaries

## Non-smooth bifurcation in nonlinear electrical circuits with multiple switching boundaries

Zhang Yin, Bi Qin-Sheng
• #### Abstract

The fast-slow dynamics of a nonlinear electrical circuit with multiple switching boundaries is investigated in this paper. For suitable parameters, periodic bursting phenomenon can be observed. The full system can be divided into slow and fast subsystems because of the difference between variational speeds of state variables. According to the slow-fast analysis, the slow variable, which modulates the behavior of the system, can be treated as a quasi-static bifurcation parameter for the fast subsystem to analyze the stabilities of equilibrium points in different areas of vector field. The bifurcation is dependent on the switching boundary in the vector field. In particular, for the two-time scale non-smooth system with fast-slow effect, the bifurcation of fast subsystem is determined by the characteristics of equilibrium points on both sides of the switching boundary. Furthermore, the generalized Jacobian matrix at the non-smooth boundary is introduced to explore the type of non-smooth bifurcation (i.e., multiple crossing bifurcation) in the fast subsystem, which can also be used to explain the mechanism for symmetric bursting phenomenon of the full system.

• Funds:

#### References

 [1] Li G L, Chen X Y 2010 Chin. Phys. B 19 030507 [2] Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2326(in Chinese) [陈章耀、张晓芳、毕勤胜 2010 物理学报 59 2326] [3] Zhang XF, Chen Z Y, Bi Q S 2009 Acta Phys. Sin. 58 2963(in Chinese) [张晓芳、陈章耀、毕勤胜 2010 物理学报 58 2963] [4] Zhang H B, Xia J W, Yu Y B, Dang C Y 2010 Chin. Phys. B 19 030505 [5] Wang F Q, Liu C X 2007 Chin. Phys. 16 942 [6] Han X J, Jiang B, Bi Q S 2009 Acta Phys. Sin. 58 111(in Chinese) [韩修静、江 波、毕勤胜 2009 物理学报 58 6006] [7] Chua L O, Lin G N 1990 IEEE Trans Circ Syst. 37 885 [8] Stouboulos I N, Miliou A N, Valaristos A P 2007 Chaos, Solutions ＆ Fractals 33 1256 [9] Koliopanos C L, Kyprianidis I M, Stouboulos I N 2003 Chaos, Solutions ＆ Fractals 16 173 [10] Ren H P, Li W C, Liu D 2010 Chin. Phys. B 19 030511 [11] Mease K D 2005 Appl. Math. Comput. 164 627 [12] Xing Z C, Xu W, Rong H W, Wang B Y 2009 Acta Phys. Sin. 58 0824(in Chinese) [邢真慈、徐 伟、戎海武、王宝燕 2009 物理学报 58 0824] [13] Yang Z Q, Lu Q S 2008 Sci. China Ser. G-Phys. Mech. Astron. 51 687 [14] Izhikevich E M 2000 International Journal of Bifurcation and Chaos 10 1171 [15] Yu S M, Qiu S S 2003 Science China E 33 365(in Chinese) [禹思敏、丘水生 2003 中国科学(E辑) 33 365] [16] Rinzel J, Ermentrout, Method in neuronal modeling ed Koch C and Segev I (Cambridge: The MIT Press) [17] Leine R I, Campen D H 2006 European Journal of Mechanics A/Solids 25 595 [18] Ji Y, Bi Q S 2010 Physics Letters A 374 1434

#### Cited By

•  [1] Li G L, Chen X Y 2010 Chin. Phys. B 19 030507 [2] Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2326(in Chinese) [陈章耀、张晓芳、毕勤胜 2010 物理学报 59 2326] [3] Zhang XF, Chen Z Y, Bi Q S 2009 Acta Phys. Sin. 58 2963(in Chinese) [张晓芳、陈章耀、毕勤胜 2010 物理学报 58 2963] [4] Zhang H B, Xia J W, Yu Y B, Dang C Y 2010 Chin. Phys. B 19 030505 [5] Wang F Q, Liu C X 2007 Chin. Phys. 16 942 [6] Han X J, Jiang B, Bi Q S 2009 Acta Phys. Sin. 58 111(in Chinese) [韩修静、江 波、毕勤胜 2009 物理学报 58 6006] [7] Chua L O, Lin G N 1990 IEEE Trans Circ Syst. 37 885 [8] Stouboulos I N, Miliou A N, Valaristos A P 2007 Chaos, Solutions ＆ Fractals 33 1256 [9] Koliopanos C L, Kyprianidis I M, Stouboulos I N 2003 Chaos, Solutions ＆ Fractals 16 173 [10] Ren H P, Li W C, Liu D 2010 Chin. Phys. B 19 030511 [11] Mease K D 2005 Appl. Math. Comput. 164 627 [12] Xing Z C, Xu W, Rong H W, Wang B Y 2009 Acta Phys. Sin. 58 0824(in Chinese) [邢真慈、徐 伟、戎海武、王宝燕 2009 物理学报 58 0824] [13] Yang Z Q, Lu Q S 2008 Sci. China Ser. G-Phys. Mech. Astron. 51 687 [14] Izhikevich E M 2000 International Journal of Bifurcation and Chaos 10 1171 [15] Yu S M, Qiu S S 2003 Science China E 33 365(in Chinese) [禹思敏、丘水生 2003 中国科学(E辑) 33 365] [16] Rinzel J, Ermentrout, Method in neuronal modeling ed Koch C and Segev I (Cambridge: The MIT Press) [17] Leine R I, Campen D H 2006 European Journal of Mechanics A/Solids 25 595 [18] Ji Y, Bi Q S 2010 Physics Letters A 374 1434
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•  Citation:
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##### Publishing process
• Received Date:  29 September 2010
• Accepted Date:  18 October 2010
• Published Online:  15 July 2011

## Non-smooth bifurcation in nonlinear electrical circuits with multiple switching boundaries

• 1. Faculty of Science, Jiangsu University, Zhenjiang 212013, China

Abstract: The fast-slow dynamics of a nonlinear electrical circuit with multiple switching boundaries is investigated in this paper. For suitable parameters, periodic bursting phenomenon can be observed. The full system can be divided into slow and fast subsystems because of the difference between variational speeds of state variables. According to the slow-fast analysis, the slow variable, which modulates the behavior of the system, can be treated as a quasi-static bifurcation parameter for the fast subsystem to analyze the stabilities of equilibrium points in different areas of vector field. The bifurcation is dependent on the switching boundary in the vector field. In particular, for the two-time scale non-smooth system with fast-slow effect, the bifurcation of fast subsystem is determined by the characteristics of equilibrium points on both sides of the switching boundary. Furthermore, the generalized Jacobian matrix at the non-smooth boundary is introduced to explore the type of non-smooth bifurcation (i.e., multiple crossing bifurcation) in the fast subsystem, which can also be used to explain the mechanism for symmetric bursting phenomenon of the full system.

Reference (18)

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