周期切换下Chen系统的振荡行为与非光滑分岔分析

余跃; 张春; 韩修静; 姜海波; 毕勤胜

doi:10.7498/aps.62.020508

摘要

研究了不同参数Chen系统之间进行周期切换时的分岔和混沌行为.基于平衡态分析, 考虑Chen系统在不同稳态解时通过周期切换连接生成的复合系统的分岔特性,得到系统的不同周期振荡行为. 在演化过程中,由于切换导致的非光滑性,复合系统不仅仅表现为两子系统动力特性的简单连接, 而且会产生各种分岔,导致诸如混沌等复杂振荡行为.通过Poincaré映射方法, 讨论了如何求周期切换系统的不动点和Floquet特征乘子.基于Floquet理论,判定系统的周期解是渐近稳定的.同时得到,随着参数变化,系统既可以由倍周期分岔序列进入混沌, 也可以由周期解经过鞍结分岔直接到达混沌.研究结果揭示了周期切换系统的非光滑分岔机理.

关键词:

Abstract

Complicated behaviors of a compound system with periodic switches between different types of Chen systems are investigated in detail. In the local analysis, the critical conditions such as fold bifurcation and Hopf bifurcation are derived to explore the bifurcations of the compound systems with different stable solutions in the two subsystems. Different types of oscillations of this switched system are observed, of which the mechanism is presented to show that the trajectories of the oscillations can be divided into several parts by the switching points, governed by the two subsystems respectively. Because of the non-smooth characteristics at the switching points, different forms of bifurcations may occur in the compound system, which may result in complicated dynamics such as chaotic oscillations, instead of the simple connections between the trajectories of the two subsystems. By the Poincaré mapping, the location of the fixed point and Floquet characteristic multiplier of switching system are discussed.With the variation of the parameter, the system can evolve into chaos via the cascading of period-doubling bifurcation. Besides, the system can evolve into chaos immediately by saddle-node bifurcations from period solutions.The non-smooth bifurcation mechanism of periodic switching system can be revealed by the research.

Keywords:

作者及机构信息

1.
江苏大学土木工程与力学学院, 镇江 212013;

2.
南通大学理学院, 南通 226019

基金项目: 国家自然科学基金(批准号: 20976075, 10972091)资助的课题.

Authors and contacts

1.
Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China;

2.
Facuity of Science, Nantong University, Nantong 226019, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 20976075, 10972091).

参考文献

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施引文献

[1]	Daniéle F P,Pascal C, Laura G 2001 Commun. Nonlinear Sci. Numer. Simulat. 16 916
[2]	Ueta T, Kawakami H 2002 International Symposium on Circuits and Systems Toskushima Japan, 2002 May 26-29 II-544
[3]	Putyrski M, Schultz C 2011 Chem. Biol. 18 1126
[4]	Victoriano C, Soledad F G, Emilio F 2012 Physica D: Nonlinear: Phenomena 241 5
[5]	Kim S C, Kim Y C, Yoon B Y, Kang M 2007 Computer Networks 51 606
[6]	Jing Z J, Yang Z Y, Jiang T 2006 Chaos, Solitons and Fractals 27 722
[7]	Santis E D 2011 Systems & Control Letters 60 807
[8]	Zhusubaliyev Z H, Mosekilde E 2008 Physics Letters A 372 2237
[9]	Kousaka T, Ueta T, Ma Y, Kawakami H 2006 Chaos, Solitons & Fractals 27 1019
[10]	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]
[11]	Andrei A, Yuliy B, Daniel L 2012 System & Control Letters 61 2
[12]	Xie G M, Wang L 2005 J. Math. Anal. Appl. 305 277
[13]	Cheng D, Guo L, Lin Y, Wang Y 2005 IEEE Transactions on Automatic Control 50 661
[14]	Matthias A M, Pascal M, Frank A 2012 Journal of Process Control 31 5
[15]	Chen Y G, Fei S M, Zhang K J, Yu L 2012 Mathematical and Computer Modelling 56 1
[16]	Zhusubaliyev Z T, Mosekilde E 2008 Physics Letters A 372 13
[17]	Ji Y, Bi Q S 2010 Acta Phys. Sin. 59 7612 (in Chinese) [季颖, 毕勤胜 2010 物理学报 59 7612]
[18]	Whiston G S 1992 Journal of Sound and Vibration 152 3
[19]	Hu H Y 1995 Journal of Sound and Vibration 187 3
[20]	Jin L, Lu Q S 2005 Acta Mechanica Sin. 37 40 (in Chinese) [金俐, 陆启韶 2005 固体力学学报 37 40
[21]	Leine R I 2006 Physica D: Nonlinear: Phenomena 223 121
[22]	Chen G, Ueta T 1999 Int. J. Bifur. Chaos 9 1465
[23]	Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2327 (in Chinese) [陈章耀, 张晓芳, 毕勤胜 2010 物理学报 59 2327]
[24]	Jiang H B, Zhang L P, Chen Z Y, Bi Q S 2012 Acta Phys. Sin. 61 080505 (in Chinese) [姜海波, 张丽萍, 陈章耀, 毕勤胜 2012 物理学报 61 080505]

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