可激发介质规范型的分叉分析：环上稳定的动力学交替可能吗？
Bifurcation analysis of a normal form for excitable media: Are stable dynamical alternans on a ring possible?
摘要：
本文给出了一维可激介质中行波规范形的分岔分析。最近提出的规范形式的唯象的理由给出的微分延迟方程的形式。规范形式表现出一个对称性保持的Hopf分支，它可能与Bogdanov-Takens点中的鞍结合并，以及对称性破缺空间非均匀的干草叉分支。本文研究了单脉冲在环中传播时的Hopf分支，通过中心流形约化，通过多尺度分析得到了波列的振幅方程，并由此得到了一个真实的Ginzburg-Landau方程。中心流形约化和多尺度分析都表明，Hopf分支总是亚临界的，与参数无关。这可能与迄今为止被认为是从超临界分叉产生的稳定振荡的心脏交替有关。我们讨论了心脏交替的影响，并重新审视一些可兴奋的介质中的振荡被认为是稳定的不稳定性。特别是，我们表明，我们的条件的发病的霍普夫分岔与众所周知的心脏交替的恢复条件相吻合。
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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