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离散动力系统1∶2共振情形下的余维二分岔反控制

杨宇娇 徐慧东 张建文

杨宇娇,徐慧东,张建文. 离散动力系统1∶2共振情形下的余维二分岔反控制 [J]. 应用数学和力学,2022,43(2):1-14 doi: 10.21656/1000-0887.420118
引用本文: 杨宇娇,徐慧东,张建文. 离散动力系统1∶2共振情形下的余维二分岔反控制 [J]. 应用数学和力学,2022,43(2):1-14 doi: 10.21656/1000-0887.420118
YANG Yujiao, XU Huidong, ZHANG Jianwen. Anti-Controlling Codimension-Two Bifurcation of Discrete Dynamical System in 1 ∶ 2 Resonance[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420118
Citation: YANG Yujiao, XU Huidong, ZHANG Jianwen. Anti-Controlling Codimension-Two Bifurcation of Discrete Dynamical System in 1 ∶ 2 Resonance[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420118

离散动力系统1∶2共振情形下的余维二分岔反控制

doi: 10.21656/1000-0887.420118
基金项目: 国家自然科学基金(11872264)
详细信息
    作者简介:

    杨宇娇(1996—),女,硕士生(E-mail: yangyujiao2333@163.com)

    张建文(1962—),男,教授,博士(通讯作者. E-mail: zhangjianwen@tyut.edu.cn)

  • 中图分类号: O193

Anti-Controlling Codimension-Two Bifurcation of Discrete Dynamical System in 1 ∶ 2 Resonance

  • 摘要: 从分岔反控制的角度设计了一套非线性反馈控制策略,来实现离散动力系统1∶2共振情形下余维二分岔的各种分岔解。首先,针对传统分岔准则在确定高余维分岔点时存在的局限性,建立了一个1∶2共振情形下的余维二分岔的新显式准则,基于这个显式准则通过设计线性控制增益来确保此类余维二分岔的存在性。然后,推导了1∶2共振的中心流形,并基于范式方法通过设计非线性控制增益,分析了1∶2共振情形下余维二分岔解的类型和稳定性。最后,以一个Arneodo-Coullet-Tresser映射为例,在指定的参数点处通过控制实现了具有1∶2共振分岔特性的各种分岔解,进一步验证了理论分析。
  • 图  1  控制参数分岔图

    Figure  1.  Control parameter bifurcation diagram

    图  2  $ \tilde{C}\left( 0,\boldsymbol{K_{NL}} \right) $的曲线图

    Figure  2.  The graph of $ \tilde{C}\left( 0,\boldsymbol{K_{NL}} \right) $

    图  3  控制系统(29)对应情形5形如式(20)的范式映射的分岔图: (a) ${\tilde{C}}(0, {{\boldsymbol{K}}_{N L}}) > 0 $时,系统(29)对应的分支图;(b) 对应于图3(a)分岔点上半部分的局部放大图;(c) 对应于图3(a)分岔点下半部分的局部放大图

    Figure  3.  Bifurcation diagram of normal form mapping like eq. (20) of control system (29) corresponding to case 5: (a) bifurcation diagram of system (29) when $\tilde{C}\left(0, {{\boldsymbol{K}}_{N L}}\right) > 0$; (b) local enlargement of the upper half part of the bifurcation point in fig. 3 (a); (c) Local enlargement of the lower half part of the bifurcation point in fig. 3 (a)

    图  4  对应于图3所示分岔图的相图:(a) 在$\left(k_{11}, k_{12}\right)={{\boldsymbol{Q}}_{1}}+(-0.08,0.15)$处的稳定点; (b) 在 $\left(k_{11}, k_{12}\right)={{\boldsymbol{Q}}_{1}}+(-0.1,-0.01)$处的不变圈; (c) 异宿分支曲线G上的异宿轨道

    Figure  4.  Phase diagram corresponding to the bifurcation diagram shown in fig. 3: (a) the fixed point at $\left(k_{11}, k_{12}\right)={{\boldsymbol{Q}}_{1}}+(-0.08,0.15)$(b) invariable circle at $\left(k_{11}, k_{12}\right)={{\boldsymbol{Q}}_{1}}+(-0.1,-0.01) $; (c) heterotropic orbit on heterotropic curve G

    图  5  控制系统(29)对应情形6形如式(20)的范式映射的分岔图:(a) $\tilde{C}\left(0, {{\boldsymbol{K}}_{N L}}\right) < 0 $ 时,系统(29)对应的分支图; (b) 对应于图5(a)分岔点上半部分的局部放大图; (c) 对应于图5(a)分岔点下半部分的局部放大图

    Figure  5.  Bifurcation diagram of normal form mapping like eq. (20) of control system (29) corresponding to case 6: (a) Bifurcation diagram of system (29) when $\tilde{C}\left(0, {{\boldsymbol{K}}_{N L}}\right) < 0 $; (b) Local enlargement of the upper half of the bifurcation point in fig. 5 (a); (c) Local enlargement of the lower half of the bifurcation point in fig. 5 (a)

    图  6  对应于图5所示分岔图的相图:(a) 在 $\left(k_{11}, k_{12}\right)=\boldsymbol{Q}_{1}+(-0.11,0.12)$ 处的稳定点; (b) 在 $\left(k_{11}, k_{12}\right)=\boldsymbol{Q}_{1}+(-0.1,-0.05)$ 处的不变圈; (c) 同宿分支曲线P上的同宿轨道; (d) 环的折分支曲线K上的轨道

    Figure  6.  Phase diagram corresponding to the bifurcation diagram shown in fig. 5: (a) the fixed point at $\left(k_{11}, k_{12}\right)=\boldsymbol{Q}_{1}+(-0.11,0.12) $; (b) Invariable circle at $\left(k_{11}, k_{12}\right)=\boldsymbol{Q}_{1}+(-0.1,-0.05) $; (c) homoclinic orbit on homoclinic curve P; (d) the orbit on the folded branch curve of the ring K

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出版历程
  • 收稿日期:  2021-04-30
  • 录用日期:  2021-04-30
  • 修回日期:  2021-06-19
  • 网络出版日期:  2022-01-08

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