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The complex dynamical evolution of a circuit system composed of two nonlinear circuit subsystems, which is switched by a periodic switching, is investigated. According to the fact that the magnification of an open-loop operational amplifier is maximum magnification, namely, the operational amplifier is always in a positive or negative saturated state, when an input voltage becomes positive from negative through zero, the output voltage jumps from the positive saturation into negative saturation. In this paper the jump function is selected as a nonlinear part in subsystems. Firstly through the stability analysis of the subsystems, their oscillation behaviors in the parameter space are given correspondingly. Secondly the complex oscillation behavior and mechanism of the switched system are discussed in the parameter space of one subsystem. The periodic orbit of the switched system is divided into four parts, influenced by non-smooth characteristics of the subsystems and switching. With the variation of the parameters, grazing bifurcation appears, and then the whole periodic orbit is separated into two symmetrical periodic oscillations. Finally the convesion of switching points into the periodic oscillation is given,and the mechanism at switching point is discussed.
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Keywords:
- jump circuit /
- switch /
- non-smooth /
- periodic oscillation
[1] Wyczalek F A 2001 IEEE Aerospace and Electronics Systems Magazine 16 15
[2] Varaiya P P 1993 IEEE Transactions on Automatic Control 38 195
[3] Dalvi A, Guay M 2009 Control Engineering Practice 17 924
[4] Yildirim H, Frank G, Bernard B 2004 Automatica 40 1647
[5] Yoshiyasu S, Masaru N, Hirokazu N, Yoshiyuki Y, Masatoshi Y, Sigeru M 2006 Computer Aided Chemical Engineering 21 1515
[6] Sun Z D, Zheng D Z 2001 IEEE Transactions on Automatic Control 46 291
[7] Leine R I 2006 Physica D 223 121
[8] Guo S Q, Yang S P, Guo J B 2005 Journal of Vibration Engineering 18 276 (in Chinese) [郭树起, 杨绍普, 郭京波 2005 振动工程学报 18 276]
[9] Li S L, Zhang Z D, Wu T Y, Bi Q S 2012 Acta Phys. Sin. 61 060504 (in Chinese) [李绍龙, 张正娣, 吴天一, 毕勤胜 2012 物理学报 61 060504]
[10] Xu X P, Antsaklis P J 2000 International Journal of Control 73 1261
[11] Hu B, Xu X, Antsaklis P J 1999 Systems & Control Letters 38 197
[12] Zhang L G, Chen Y Z, Cui P Y 2005 Nonlinear Analysis 62 1527
[13] Cheng D Z 2003 Systems & Control Letters 51 79
[14] Wu T Y, Zhang Z D, Bi Q S 2012 Acta Phys. Sin. 61 070502 (in Chinese) [吴天一, 张正娣, 毕勤胜 2012 物理学报 61 070502]
[15] Lu T X 2002 Overview of Nonlinear Physics (Hefei: China University of Science and Technology Press) p173 (in Chinese) [陆同兴 2002 非线性物理概论(合肥:中国科技大学出版社) 第173页]
[16] Contou-Carrere M N, Daoutidis P 2005 IEEE Transactions on Automatic Control 50 1831
[17] Sprott J C 2000 Physics Letters A 266 19
[18] Galvenetto U 2001 Journal of Sound and Vibration 248 653
[19] Zhang X G, Ma Y D, Li S L 2011 Nonlinear Circuit: Basic Analysis and Design (Beijing: Higher Education Press) p18 (in Chinese) [张新国, 马义德, 李守亮 2011 非线性电路:基础分析与设计(北京:高等教育出版社) 第18页]
[20] Gao W H, Guo X, Jiang J 2011 Journal of Dynamics and Control 9 363 (in Chinese) [高文辉, 郭旭, 江俊 2011 动力学与控制学报 9 363]
[21] Qin Z Y, Lu Q S 2009 Journal of Vibration and Shock 28 80 (in Chinese) [秦志英, 陆启韶 2009 振动与冲击 28 80]
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[1] Wyczalek F A 2001 IEEE Aerospace and Electronics Systems Magazine 16 15
[2] Varaiya P P 1993 IEEE Transactions on Automatic Control 38 195
[3] Dalvi A, Guay M 2009 Control Engineering Practice 17 924
[4] Yildirim H, Frank G, Bernard B 2004 Automatica 40 1647
[5] Yoshiyasu S, Masaru N, Hirokazu N, Yoshiyuki Y, Masatoshi Y, Sigeru M 2006 Computer Aided Chemical Engineering 21 1515
[6] Sun Z D, Zheng D Z 2001 IEEE Transactions on Automatic Control 46 291
[7] Leine R I 2006 Physica D 223 121
[8] Guo S Q, Yang S P, Guo J B 2005 Journal of Vibration Engineering 18 276 (in Chinese) [郭树起, 杨绍普, 郭京波 2005 振动工程学报 18 276]
[9] Li S L, Zhang Z D, Wu T Y, Bi Q S 2012 Acta Phys. Sin. 61 060504 (in Chinese) [李绍龙, 张正娣, 吴天一, 毕勤胜 2012 物理学报 61 060504]
[10] Xu X P, Antsaklis P J 2000 International Journal of Control 73 1261
[11] Hu B, Xu X, Antsaklis P J 1999 Systems & Control Letters 38 197
[12] Zhang L G, Chen Y Z, Cui P Y 2005 Nonlinear Analysis 62 1527
[13] Cheng D Z 2003 Systems & Control Letters 51 79
[14] Wu T Y, Zhang Z D, Bi Q S 2012 Acta Phys. Sin. 61 070502 (in Chinese) [吴天一, 张正娣, 毕勤胜 2012 物理学报 61 070502]
[15] Lu T X 2002 Overview of Nonlinear Physics (Hefei: China University of Science and Technology Press) p173 (in Chinese) [陆同兴 2002 非线性物理概论(合肥:中国科技大学出版社) 第173页]
[16] Contou-Carrere M N, Daoutidis P 2005 IEEE Transactions on Automatic Control 50 1831
[17] Sprott J C 2000 Physics Letters A 266 19
[18] Galvenetto U 2001 Journal of Sound and Vibration 248 653
[19] Zhang X G, Ma Y D, Li S L 2011 Nonlinear Circuit: Basic Analysis and Design (Beijing: Higher Education Press) p18 (in Chinese) [张新国, 马义德, 李守亮 2011 非线性电路:基础分析与设计(北京:高等教育出版社) 第18页]
[20] Gao W H, Guo X, Jiang J 2011 Journal of Dynamics and Control 9 363 (in Chinese) [高文辉, 郭旭, 江俊 2011 动力学与控制学报 9 363]
[21] Qin Z Y, Lu Q S 2009 Journal of Vibration and Shock 28 80 (in Chinese) [秦志英, 陆启韶 2009 振动与冲击 28 80]
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