Volume 43 Issue 2
Feb.  2022
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YANG Yujiao, XU Huidong, ZHANG Jianwen. Anti-Controlling Codimension-2 Bifurcation of Discrete Dynamical Systems in 1 ∶ 2 Resonance[J]. Applied Mathematics and Mechanics, 2022, 43(2): 142-155. doi: 10.21656/1000-0887.420118
Citation: YANG Yujiao, XU Huidong, ZHANG Jianwen. Anti-Controlling Codimension-2 Bifurcation of Discrete Dynamical Systems in 1 ∶ 2 Resonance[J]. Applied Mathematics and Mechanics, 2022, 43(2): 142-155. doi: 10.21656/1000-0887.420118

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

doi: 10.21656/1000-0887.420118
  • Received Date: 2021-04-30
  • Accepted Date: 2021-04-30
  • Rev Recd Date: 2021-06-19
  • Available Online: 2022-01-08
  • Publish Date: 2022-02-01
  • 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 in 1∶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.

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