Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand.
Philos Trans A Math Phys Eng Sci. 2013 Aug 19;371(1999):20120467. doi: 10.1098/rsta.2012.0467. Print 2013 Sep 28.
Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the event that it bifurcates unstably. The approach is an extension of the Pyragas delayed feedback method, as it was developed for the generic subcritical Hopf bifurcation problem. Restrictions on the types of groups where the proposed method works are given. After addition of the appropriately optimized feedback term, we are able to compute the stability of the targeted solution using standard bifurcation theory, and give an account of the parameter regimes in which stabilization is possible. We conclude by demonstrating our results with a numerical example involving symmetrically coupled identical nonlinear oscillators.
对称破缺 Hopf 分岔问题在模式形成的研究中自然出现。这些等变 Hopf 分岔通常会导致多个解分支从完全对称的平衡态同时分叉。等变 Hopf 分岔定理根据它们的对称性对这些解分支进行分类，对称性可能涉及空间变换和时间推移的组合。在本文中，我们利用这些时空对称性设计非侵入性反馈控制，以选择和稳定目标解分支，以防其不稳定分岔。该方法是 Pyragas 延迟反馈方法的扩展，因为它是为一般亚临界 Hopf 分岔问题开发的。给出了所提出的方法适用的群类型的限制。在添加适当优化的反馈项后，我们能够使用标准分岔理论计算目标解的稳定性，并说明稳定化的参数范围。最后，我们通过一个涉及对称耦合相同非线性振荡器的数值例子展示了我们的结果。
