二维正弦离散映射的分岔和吸引子

毕闯; 张千; 向勇; 王京梅

doi:10.7498/aps.62.240503

摘要

由一个正弦映射和一个三次方映射通过非线性耦合，构成一个新的二维正弦离散映射. 基于此二维正弦离散映射得到系统的不动点以及相应的特征值，分析了系统的稳定性，研究了系统的复杂非线性动力学行为及其吸引子的演变过程. 研究结果表明：此二维正弦离散映射中存在复杂的对称性破缺分岔、Hopf分岔、倍周期分岔和周期振荡快慢效应等非线性物理现象. 进一步根据控制变量变化时系统的分岔图、Lyapunov指数图和相轨迹图分析了系统的分岔模式共存、快慢周期振荡及其吸引子的演变过程，通过数值仿真验证了理论分析的正确性.

关键词:

Abstract

A new two-dimensional sinusoidal discrete map is achieved by nonlinearly coupling a sinusoidal map and with a cubic map. The fixed points and the corresponding eigenvalues are obtained based on this two-dimensional sinusoidal discrete map, and the stability of the system is analyzed to study the complex nonlinear dynamic behavior of the system and the evolutions of their attractors. The research results indicate that there are complex nonlinear physical phenomena in this two-dimensional sinusoidal discrete map, such as symmetry breaking bifurcation, Hopf bifurcation, period doubling bifurcation, periodic oscillation fast-slow effect, etc. Furthermore, bifurcation mode coexisting, fast-slow periodic oscillations and the evolutions of the attractors of the system are analyzed by using the bifurcation diagram, the Lyapunov exponent diagram and the phase portraits when the control parameters of the system are varied, and the correctness of the theoretical analysis is verified based on numerical simulations.

Keywords:

作者及机构信息

1.
电子科技大学, 电子薄膜与集成器件国家重点实验室, 成都 610054;

2.
电子科技大学能源科学与工程学院, 成都 611731

基金项目: 电子薄膜与集成器件国家重点实验室基础研究开放创新基金（批准号：CXJJ2010001）资助的课题.

Authors and contacts

1.
State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China;

2.
School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

Funds: Project supported by the Open Innovation Fund for Basic Research of State Key Laboratory of Electronic Thin Films and Integrated Devices, China (Grant No. CXJJ2010001).

参考文献

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[2]	Zhang H G, Fu J, Ma T D, Tong S C 2009 Chin. Phys. B 18 969
[3]	Parui S, Banerjee S 2003 IEEE Trans. Circuits Syst. I 50 1464
[4]	Yan H, Wei P, Xiao X C 2010 Chin. Phys. B 19 090501
[5]	Yan H, Wei P, Xiao X C 2009 Chin. Phys. B 18 3287
[6]	Van Ha N, Han J S 2013 Chin. Phys. Lett. 30 060501
[7]	Wu J, Zhan X S, Zhang X H, Gao H L 2012 Chin. Phys. Lett. 29 050203
[8]	Namajunas A, Pyragas K, Tamasevicius A 1997 Int. J. Bifurc. Chaos 7 957
[9]	Wu X, Tse C K, Dranga O, Lu J 2006 IEEE Trans. Circuits Syst. I 53 204
[10]	Yang R, Zhang B 2007 Chin. Soc. Elec. Eng. 27 114 (in Chinese) [杨汝, 张波 2007 中国电机工程学报 27 114]
[11]	Xu J, Long K P, Fournier-Prunaret D, Taha A K, Charge P 2010 Chin. Phys. Lett. 27 020504
[12]	Xu J, Long K P, Fournier-Prunaret D, Taha A K, Charge P 2010 Chin. Phys. Lett. 27 080506
[13]	Boyland P L 1986 Comm. Math. Phys. 106 353
[14]	Lalescu C C 2010 Arxiv preprint arXiv 1011 6552
[15]	Hu W, Zhao G H, Zhang G, Zhang J Q, Liu X L 2012 Acta Phys. Sin. 61 170505 (in Chinese) [胡文, 赵广浩, 张弓, 张景乔, 刘贤龙 2012 物理学报 61 170505]
[16]	Zhang H, Chu Y D, Ding W C, Li X F 2013 Acta Phys. Sin. 62 040202 (in Chinese) [张惠, 褚衍东, 丁旺才, 李险峰 2013 物理学报 62 040202]
[17]	Bao B C, Kang Z S, Xu J P, Hu W 2009 Acta Phys. Sin. 58 1420 (in Chinese) [包伯成, 康祝圣, 许建平, 胡文 2009 物理学报 58 1420]
[18]	Meng J D, Bao B C, Xu Q 2011 Acta Phys. Sin. 60 010504 (in Chinese) [孟继德, 包伯成, 徐强 2011 物理学报 60 010504]
[19]	Edward O 2002 Chaos in Dynamical System (Cambridge: Cambridge University Press) p50
[20]	Zhou G H, Xu J P, Bao B C, Zhang F, Liu X S 2010 Chin. Phys. Lett. 27 090504

施引文献

[1]	Ma T D, Fu J, Sun Y 2010 Chin. Phys. B 19 090502
[2]	Zhang H G, Fu J, Ma T D, Tong S C 2009 Chin. Phys. B 18 969
[3]	Parui S, Banerjee S 2003 IEEE Trans. Circuits Syst. I 50 1464
[4]	Yan H, Wei P, Xiao X C 2010 Chin. Phys. B 19 090501
[5]	Yan H, Wei P, Xiao X C 2009 Chin. Phys. B 18 3287
[6]	Van Ha N, Han J S 2013 Chin. Phys. Lett. 30 060501
[7]	Wu J, Zhan X S, Zhang X H, Gao H L 2012 Chin. Phys. Lett. 29 050203
[8]	Namajunas A, Pyragas K, Tamasevicius A 1997 Int. J. Bifurc. Chaos 7 957
[9]	Wu X, Tse C K, Dranga O, Lu J 2006 IEEE Trans. Circuits Syst. I 53 204
[10]	Yang R, Zhang B 2007 Chin. Soc. Elec. Eng. 27 114 (in Chinese) [杨汝, 张波 2007 中国电机工程学报 27 114]
[11]	Xu J, Long K P, Fournier-Prunaret D, Taha A K, Charge P 2010 Chin. Phys. Lett. 27 020504
[12]	Xu J, Long K P, Fournier-Prunaret D, Taha A K, Charge P 2010 Chin. Phys. Lett. 27 080506
[13]	Boyland P L 1986 Comm. Math. Phys. 106 353
[14]	Lalescu C C 2010 Arxiv preprint arXiv 1011 6552
[15]	Hu W, Zhao G H, Zhang G, Zhang J Q, Liu X L 2012 Acta Phys. Sin. 61 170505 (in Chinese) [胡文, 赵广浩, 张弓, 张景乔, 刘贤龙 2012 物理学报 61 170505]
[16]	Zhang H, Chu Y D, Ding W C, Li X F 2013 Acta Phys. Sin. 62 040202 (in Chinese) [张惠, 褚衍东, 丁旺才, 李险峰 2013 物理学报 62 040202]
[17]	Bao B C, Kang Z S, Xu J P, Hu W 2009 Acta Phys. Sin. 58 1420 (in Chinese) [包伯成, 康祝圣, 许建平, 胡文 2009 物理学报 58 1420]
[18]	Meng J D, Bao B C, Xu Q 2011 Acta Phys. Sin. 60 010504 (in Chinese) [孟继德, 包伯成, 徐强 2011 物理学报 60 010504]
[19]	Edward O 2002 Chaos in Dynamical System (Cambridge: Cambridge University Press) p50
[20]	Zhou G H, Xu J P, Bao B C, Zhang F, Liu X S 2010 Chin. Phys. Lett. 27 090504

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