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摘要翻译：
本文给出了一维可激发介质中行波规范形的分岔分析。本文以微分时滞方程的形式给出了最近在唯象基础上提出的规范形。该范式表现出在Bogdanov-Takens点上可能与鞍结点合并的保持对称性的Hopf分岔和对称性破缺的空间非均匀干草叉分岔。本文用中心流形约化方法研究了单脉冲在环中传播的Hopf分岔，用多尺度分析方法研究了波列传播的Hopf分岔，得到了相应的振幅方程&实Ginzburg-Landau方程。中心流形约化和多尺度分析都表明Hopf分岔总是亚临界的，与参数无关。这可能与心脏交替有联系，到目前为止，心脏交替被认为是由超临界分叉发出的稳定振荡。我们讨论了对心脏交替的影响，并重新讨论了在一些被认为振荡稳定的可兴奋介质中的不稳定性。特别地，我们证明了我们的Hopf分叉开始的条件与众所周知的心脏交替的恢复条件是一致的。
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英文标题：
《Bifurcation analysis of a normal form for excitable media: Are stable
dynamical alternans on a ring possible?》
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作者：
G.A. Gottwald
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最新提交年份：
2008
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分类信息：
一级分类：Physics 物理学
二级分类：Pattern Formation and Solitons 图形形成与孤子
分类描述：Pattern formation, coherent structures, solitons
图案形成，相干结构，孤子
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一级分类：Quantitative Biology 数量生物学
二级分类：Other Quantitative Biology 其他定量生物学
分类描述：Work in quantitative biology that does not fit into the other q-bio classifications
不适合其他q-bio分类的定量生物学工作
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英文摘要：
We present a bifurcation analysis of a normal form for travelling waves in one-dimensional excitable media. The normal form which has been recently proposed on phenomenological grounds is given in form of a differential delay equation. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with a saddle-node in a Bogdanov-Takens point, and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both, the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.
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PDF链接：
https://arxiv.org/pdf/0802.1750
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