具有多分界面的非线性电路中的非光滑分岔

张银; 毕勤胜

doi:10.7498/aps.60.070507

摘要

本文分析了具有多分界面的非线性电路在不同时间尺度下的快慢动力学行为. 在一定的参数条件下,系统的周期解为簇发解,表现出明显的快慢效应. 根据状态变量变化的快慢,把全系统划分为快子系统和慢子系统两组. 根据快慢分析法将慢变量看作快子系统的控制参数,分析了快子系统的平衡点在向量场不同区域内的稳定性. 非光滑系统的分岔与向量场的分界面密切相关,对于具有快慢效应的两时间尺度非光滑系统,快子系统的分岔则取决于分界面两侧平衡点的性质. 通过在临界面引入广义Jacobi矩阵,讨论了快子系统非光滑分岔的类型,即多次穿越分

关键词:

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.

Keywords:

作者及机构信息

张银,
毕勤胜

1.
江苏大学理学院,镇江 212013

基金项目: 国家自然科学基金(批准号:10972091和10872080)和江苏大学高级人才基金(批准号:10JDG062)资助的课题.

Authors and contacts

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

参考文献

[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

施引文献

[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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