二维抛物线离散映射的动力学研究

孟继德; 包伯成; 徐强

doi:10.7498/aps.60.010504

摘要

由两个一维抛物线离散映射作推广并非线性耦合,实现了一个新的二维抛物线离散映射.利用不动点稳定性分析和映射分岔分析,研究了所提出的二维离散映射的复杂动力学行为及其吸引子的演变过程,阐述了它所特有的共存分岔模式和快慢周期振荡效应等动力学特性.研究结果表明:二维抛物线离散映射具有动力学特性调节和动态幅度调节的两个功能不同的控制参数,存在Hopf分岔、分岔模式共存、锁频和周期振荡快慢效应等非线性物理现象.并基于微控制器实现的数字电路验证了相应的理论分析和数值仿真结果.

关键词:

二维离散映射 /
分岔 /
吸引子 /
参数

Abstract

By extending and nonlinearly coupling two one-dimensional parabolic discrete maps, a new two-dimensional parabolic discrete map is achieved. By using stability analysis of fixed points and bifurcation analysis of map, the complex dynamical behavior and attractor evolution of the proposed two-dimensional discrete map are investigated, and its peculiar dynamical characteristics, such as the coexisting bifurcation modes and fast-slow periodic oscillation effects, etc., are illustrated. The research results indicate that two-dimensional parabola discrete map has two control parameters with different functions of adjustable dynamical behaviors and adjustable dynamic amplitudes, and there emerge nonlinear physical phenomena of Hopf bifurcation, bifurcation mode coexisting, locked-frequency and periodic oscillation fast-slow effect. Furthermore, the corresponding theoretical analysis and numerical simulation results are verified based on a digital circuit realized by microcontroller.

Keywords:

作者及机构信息

(1)常州工学院计算机信息工程学院,常州 213002; (2)江苏技术师范学电气信息工程学院,常州 213001

基金项目: 江苏省自然科学基金(批准号:BK2009105)资助的课题.

Authors and contacts

(1)School of Computer and Information Engineering, Changzhou Institute of Technology, Changzhou 213002, China; (2)School of Electrical and Information Engineering, Jiangsu Teachers University of Technology, Changzhou 213001, China

参考文献

[1]	Liu Z, Zhu X H, Hu W, Jiang F 2007 Int. J. Bifur. Chaos 17 1735
[2]	Lin F Y, Liu J M 2004 IEEE J. Quantum Electron. 40 815
[3]	Long M, Qiu S S 2007 Chin. Phys. 16 2254
[4]	Bao B C, Xu J P, Liu Z 2009 Chin. Phys. B 18 4742
[5]	Zheng F, Tian X J, Li X Y, Wu B 2008 Chin. Phys. B 17 1685
[6]	May R M 1976 Nature 261 459
[7]	Feigenbaum M J 1978 J. Statistical Phys. 19 25
[8]	Yu W P, Wei X P 2006 Acta Phys. Sin. 55 3969 (in Chinese) [于万波、魏小鹏 2006 物理学报 55 3969]
[9]	Bao B C, Hu W, Liu Z, Kang Z S, Xu J P 2009 Acta Phys. Sin. 58 2440 (in Chinese) [包伯成、胡文、刘中、康祝圣、许建平 2009 物理学报 58 2440]
[10]	Bao B C, Xu J P, Liu Z 2009 Acta Phys. Sin. 58 2949 (in Chinese) [包伯成、许建平、刘中 2009 物理学报 58 2949]
[11]	Wang X Y, Liang Q Y 2005 Chin. J. Theoretical and Applied Mechanics 37 522 (in Chinese) [王兴元、梁庆永 2005 力学学报 37 522]
[12]	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]
[13]	Wang X Y, Luo C 2005 Chin. J. Theoretical and Applied Mechanics 37 346 (in Chinese) [王兴元、骆超 2005 力学学报 37 346]
[14]	Wang X Y, Wang M J 2008 Acta Phys. Sin. 57 731 (in Chinese) [王兴元、王明军 2008 物理学报 57 731]
[15]	Wang L M 2009 J. Northwest Normal Univ. (Natural Science) 45 26 (in Chinese) [王立军 2009 西北师范大学学报(自然科学版) 45 26]
[16]	Wang X Y, Luo C 2006 J. Software 17 729
[17]	Li Z G, Xu D L 2004 Chaos, Solitons and Fractals 22 477
[18]	Zhang X F, Chen Z Y, Ji Y, Bi Q S 2009 Chin. J. Theoretical and Applied Mechanics 41 765 (in Chinese) [张晓芳、陈章耀、季颖、毕勤胜 2009 力学学报 41 765]
[19]	Li C B, Wang D C 2009 Acta Phys. Sin. 58 764 (in Chinese) [李春彪、王德纯 2009 物理学报 58 764]
[20]	Li C B, Wang H K, Chen S 2010 Acta Phys. Sin. 59 783 (in Chinese) [李春彪、王翰康、陈谡 2010 物理学报 59 783]

施引文献

[1]	Liu Z, Zhu X H, Hu W, Jiang F 2007 Int. J. Bifur. Chaos 17 1735
[2]	Lin F Y, Liu J M 2004 IEEE J. Quantum Electron. 40 815
[3]	Long M, Qiu S S 2007 Chin. Phys. 16 2254
[4]	Bao B C, Xu J P, Liu Z 2009 Chin. Phys. B 18 4742
[5]	Zheng F, Tian X J, Li X Y, Wu B 2008 Chin. Phys. B 17 1685
[6]	May R M 1976 Nature 261 459
[7]	Feigenbaum M J 1978 J. Statistical Phys. 19 25
[8]	Yu W P, Wei X P 2006 Acta Phys. Sin. 55 3969 (in Chinese) [于万波、魏小鹏 2006 物理学报 55 3969]
[9]	Bao B C, Hu W, Liu Z, Kang Z S, Xu J P 2009 Acta Phys. Sin. 58 2440 (in Chinese) [包伯成、胡文、刘中、康祝圣、许建平 2009 物理学报 58 2440]
[10]	Bao B C, Xu J P, Liu Z 2009 Acta Phys. Sin. 58 2949 (in Chinese) [包伯成、许建平、刘中 2009 物理学报 58 2949]
[11]	Wang X Y, Liang Q Y 2005 Chin. J. Theoretical and Applied Mechanics 37 522 (in Chinese) [王兴元、梁庆永 2005 力学学报 37 522]
[12]	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]
[13]	Wang X Y, Luo C 2005 Chin. J. Theoretical and Applied Mechanics 37 346 (in Chinese) [王兴元、骆超 2005 力学学报 37 346]
[14]	Wang X Y, Wang M J 2008 Acta Phys. Sin. 57 731 (in Chinese) [王兴元、王明军 2008 物理学报 57 731]
[15]	Wang L M 2009 J. Northwest Normal Univ. (Natural Science) 45 26 (in Chinese) [王立军 2009 西北师范大学学报(自然科学版) 45 26]
[16]	Wang X Y, Luo C 2006 J. Software 17 729
[17]	Li Z G, Xu D L 2004 Chaos, Solitons and Fractals 22 477
[18]	Zhang X F, Chen Z Y, Ji Y, Bi Q S 2009 Chin. J. Theoretical and Applied Mechanics 41 765 (in Chinese) [张晓芳、陈章耀、季颖、毕勤胜 2009 力学学报 41 765]
[19]	Li C B, Wang D C 2009 Acta Phys. Sin. 58 764 (in Chinese) [李春彪、王德纯 2009 物理学报 58 764]
[20]	Li C B, Wang H K, Chen S 2010 Acta Phys. Sin. 59 783 (in Chinese) [李春彪、王翰康、陈谡 2010 物理学报 59 783]

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