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Equivariant degree and global Hopf bifurcation for NFDEs with symmetry.

机译：具有对称性的NFDE的等变度和全局Hopf分支。

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摘要

This thesis is concerned with the equivariant topological degree and its applications to global Hopf bifurcation theory for (neutral) functional differential equations (NFDEs) with symmetry. Firstly, an alternative new definition of the G-degree developed recently by K. Geba, W. Krawcewicz and J. Wu is carried out by using equivariant generic approximations and an analytic formula for the computation of the degree is provided. Secondly, by extending the G-degree to that for equivariant condensing field in Banach G-spaces, a local symmetric bifurcation theorem of Krasnosel'skii type and a global symmetric bifurcation theorem of Rabinowitz type for a class of composite coincidence equations with symmetry are obtained via a G-degree approach. Thirdly, the above bifurcation results are applied to prove local and global Hopf bifurcation theorems for (neutral) functional differential equations with symmetry, which include several important theorems in bifurcation theory obtained by S. N. Chow, B. Fiedler, J. Ize, J. Mallet-Paret, R. D. Nussbaum and J. Yorke. Their applications to the Rashevsky-Turing theory are also considered. Finally, the discrete global bifurcation waves such as phase-locked oscillations and synchronous oscillations in a single-species ring patch model as well as in a ring array of coupled lossless transmission lines are investigated via global symmetric Hopf bifurcation theorems.
机译：本文研究的是等变拓扑度及其在具有对称性的（中性）泛函微分方程（NFDE）的全局Hopf分岔理论中的应用。首先，使用等变类属近似对K. Geba，W。Krawcewicz和J. Wu最近开发的G度进行了新定义，并提供了用于计算度的解析公式。其次，通过将G度扩展到Banach G空间中的等变凝聚场的G度，得到一类具有对称性的复合重合方程的Krasnosel'skii型局部对称分支定理和Rabinowitz型全局对称分支定理通过G度方法。第三，上述分叉结果被用于证明具有对称性的（中性）泛函微分方程的局部和全局Hopf分支定理，其中包括SN Chow，B. Fiedler，J. Ize，J. Mallet获得的分叉理论中的几个重要定理。 -Paret，RD Nussbaum和J. Yorke。还考虑了它们在Rashevsky-Turing理论中的应用。最后，通过全局对称Hopf分叉定理，研究了单物种环形补丁模型以及耦合的无损传输线的环形阵列中的离散全局分叉波，例如锁相振荡和同步振荡。

著录项

  • 作者

    Xia, Huaxing.;

  • 作者单位

    University of Alberta (Canada).;

  • 授予单位 University of Alberta (Canada).;
  • 学科 Mathematics.;Engineering Electronics and Electrical.
  • 学位 Ph.D.
  • 年度 1994
  • 页码 285 p.
  • 总页数 285
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 老年病学;
  • 关键词

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