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一类二维非自治离散系统中的复杂簇发振荡结构

陈振阳 韩修静 毕勤胜

陈振阳, 韩修静, 毕勤胜. 一类二维非自治离散系统中的复杂簇发振荡结构[J]. 力学学报, 2017, 49(1): 165-174. doi: 10.6052/0459-1879-16-267
引用本文: 陈振阳, 韩修静, 毕勤胜. 一类二维非自治离散系统中的复杂簇发振荡结构[J]. 力学学报, 2017, 49(1): 165-174. doi: 10.6052/0459-1879-16-267
Chen Zhenyang, Han Xiujing, Bi Qinsheng. COMPLEX BURSTING OSCILLATION STRUCTURES IN A TWO-DIMENSIONAL NON-AUTONOMOUS DISCRETE SYSTEM[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 165-174. doi: 10.6052/0459-1879-16-267
Citation: Chen Zhenyang, Han Xiujing, Bi Qinsheng. COMPLEX BURSTING OSCILLATION STRUCTURES IN A TWO-DIMENSIONAL NON-AUTONOMOUS DISCRETE SYSTEM[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 165-174. doi: 10.6052/0459-1879-16-267

一类二维非自治离散系统中的复杂簇发振荡结构

doi: 10.6052/0459-1879-16-267
基金项目: 

国家自然科学基金 11572141,11632008,11502091,11472115,11402226

详细信息
    通讯作者:

    韩修静,E-mail:xjhan@mail.ujs.edu.cn

  • 中图分类号: O322

COMPLEX BURSTING OSCILLATION STRUCTURES IN A TWO-DIMENSIONAL NON-AUTONOMOUS DISCRETE SYSTEM

  • 摘要: 簇发振荡是多时间尺度系统复杂动力学行为的典型代表,簇发振荡的动力学机制与分类问题是簇发研究的重要问题之一,但当前学者们所揭示的簇发振荡的结构大多较为简单.研究以非自治离散Duffing系统为例,探讨具有复杂分岔结构的新型簇发振荡模式,并将其分为两大类,一类经由Fold分岔所诱发的对称式簇发,另一类经由延迟倍周期分岔所诱发的非对称式簇发.快子系统的分岔表现为典型的含有两个Fold分岔点的S形不动点曲线,其上、下稳定支可经由倍周期(即Flip)分岔通向混沌.当非自治项(即慢变量)穿越Fold分岔点时,系统的轨线可以向上、下稳定支的各种吸引子(例如,周期轨道和混沌)进行转迁,因此得到了经由Fold分岔所诱发的各种对称式簇发;而当非自治项无法穿越Fold分岔点,但可以穿越Flip分岔点时,系统产生了延迟Flip分岔现象.基于此,得到了经由延迟Flip分岔所诱发的各种非对称簇发.特别地,文中所报道的簇发振荡模式展现出复杂的反向Flip分岔结构.研究结果表明,这与非自治项缓慢地反向穿越快子系统的Flip分岔点有关.研究结果丰富了离散系统簇发的动力学机理和分类.

     

  • 图  1  快子系统关于单参数$\beta $的分岔图

    Figure  1.  One parameter bifurcation diagram of the fast sub-system with respect to $\beta $

    图  2  快子系统在($\beta,b$)平面上的分岔集

    Figure  2.  Bifurcation set of the fast system on the plane ($\beta ,b$)

    图  3  映射(2)关于参数 $\beta $ 的典型单参数分岔图

    Figure  3.  Typical one parameter bifurcation diagrams of the map (2) with respect to $\beta $

    图  4  不同的对称簇发

    Figure  4.  Different symmetric bursting

    图  5  对称式"Fold/级联反向 Flip"簇发,$b =0.215$,$Z_n =0.4\cos(0.001 n)$

    Figure  5.  Symmetric "Fold/a cascade of inverse Flip" bursting,$b =0.215$,$Z_n =0.4\cos(0.001 n)$

    图  6  另一种混沌簇发,$a =1.975$,$b =0$,$Z_n = 0.38 \cos(0.001 n) $

    Figure  6.  Another chaotic bursting,where $a =1.975$,$b =0$,$Z_n = 0.38\cos(0.001 n ) $

    图  7  $ b =0.2$时快子系统关于 $\beta $ 的分岔图

    Figure  7.  Bifurcation diagram of fast subsystem with $\beta $ for $b =0.2$

    图  8  "延迟Flip/反向Flip"簇发,$b =0.2$,$Z_n =0.27\cos(0.001 n)$

    Figure  8.  "Delayed F lip/inverse Flip" bursting,$b =0.2$,$Z_n =0.27\cos(0.001n)$

    图  9  "双重延迟 Flip/双重反向 Flip"簇发,$ b =0.2$,$Z_n =0.34\cos(0.001 n)$

    Figure  9.  "Delayed double Flip/double inverse Flip" bursting,$b =0.2$,$Z_n =0.34\cos(0.001 n)$

    图  10  "多重延迟 Flip/多重反向Flip"簇发,$ b =0.2$,$Z_n =0.35\cos(0.001 n)$

    Figure  10.  "Delayed multiple Flip/inverse multiple Flip" bursting,$b =0.2$,$Z_n =0.35\cos(0.001 n)$

    图  11  "级联延迟Flip /级联反向 Flip"簇发,$ b =0.2$,$Z_n =0.38\cos(0.001 n)$

    Figure  11.  "A cascade of delayed Flip /a cascade of inverse Flip" bursting, $b =0.2$,$Z_n =0.38\cos(0.001 n)$

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出版历程
  • 收稿日期:  2016-09-22
  • 修回日期:  2016-11-06
  • 网络出版日期:  2016-11-11
  • 刊出日期:  2017-01-18

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