SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 252-303, January 2020.
In this paper, we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf, and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models.

中文翻译：

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从余阶时滞微分方程平衡的两个分岔切换到非双曲周期。

SIAM应用动力系统杂志，第19卷，第1期，第252-303页，2020年1月。
在本文中，我们在延迟微分方程（DDE）中的广义Hopf（Bautin），fold-Hopf，Hopf-Hopf和跨临界-Hopf分叉附近执行参数相关的中心流形约简。这使我们能够初始化余维一均衡的继续，并循环从这些余维两个分叉点产生的分支。使用基于Fredholm备选方案的归一化技术，在对偶半群的功能分析扰动框架（太阳星演算）中得出法线形式系数。所获得的表达式给出了明确的公式，这些公式已在免费提供的数字软件包DDE-BifTool中实现。虽然我们的理论结果被证明可以更广泛地应用，软件实现和示例着重于具有有限多个离散延迟的DDE。连同DDE-BifTool的延续功能，这提供了一个强大的工具来研究此类DDE接近平衡的动力学。在各种模型上都证明了有效性。