离散动力系统1∶2共振情形下的余维二分岔反控制  

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

作　　者：杨宇娇 徐慧东 张建文 YANG Yujiao;XU Huidong;ZHANG Jianwen(College of Mathematics,Taiyuan University of Technology,Taiyuan 030024,P.R.China;College of Mechanical and Vehicle Engineering,Taiyuan University of Technology,Taiyuan 030024,P.R.China)

机构地区：[1]太原理工大学数学学院,太原030024 [2]太原理工大学机械与运载工程学院,太原030024

出　　处：《应用数学和力学》2022年第2期142-155,共14页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(11872264)。

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.A set of nonlinear feedback control strategies were designed to realize the bifurcation solutions of codimensional bifurcations in discrete dynamical systems with 1∶2 resonance from the perspective of bifurcation anti-controlling. Firstly,aimed at the limitation of traditional bifurcation criteria for determination of high codimensional bifurcation points, a new explicit criterion for codimension-2 bifurcation in 1∶2 resonance was proposed. Based on this explicit criterion, the linear control gain was designed to ensure the existence of such codimension-2 bifurcation. Then, the central manifold of 1∶2 resonance was derived. Based on the normal form method, the types and stability of codimension-2 bifurcation solutions in1∶2 resonance were analyzed through design of nonlinear control gain. Finally, an Arneodo-Coullet-Tresser mapping was taken as an example, and various bifurcation solutions with 1∶2 resonance bifurcation properties were realized by control at the specified parameter points, to further validate the theoretical analysis.

关 键 词：1∶2共振情形下的余维二分岔 显式准则 分岔反控制 离散动力系统 

分 类 号：O193[理学—数学]