论文标题
耦合Fitzhugh-Nagumo振荡器中的混合模式振荡：刺激分析
Mixed-mode oscillations in coupled FitzHugh-Nagumo oscillators: blowup analysis of cusped singularities
论文作者
论文摘要
在本文中，我们使用几何奇异扰动理论和爆炸作为我们的主要技术工具，以研究具有对称和排斥耦合的两个耦合Fitzhugh-Nagumo单元中发生的混合模式振荡（MMO）。特别是，我们证明了该模型中的MMO不是由于通用的折叠奇异性，而是由于临界歧管的尖端（而非折叠）所致。使用爆炸，我们在分析上确定SAO的数量，表明 - 至于折叠节点 - 它们是由Weber方程和特征值之比确定的。我们还表明，该模型在降低的减少问题中经历了（对称的）马鞍节点分叉，尽管在此级别上类似于折叠的马鞍节点（II型），但也发生在尖口上，而不是折叠。我们证明，这种分叉与不变缸的出现，SAOS的发作以及振幅增加的SAO有关。我们将发现与数值计算联系起来，并找到了极好的一致性。
In this paper, we use geometric singular perturbation theory and blowup, as our main technical tool, to study the mixed-mode oscillations (MMOs) that occur in two coupled FitzHugh-Nagumo units with symmetric and repulsive coupling. In particular, we demonstrate that the MMOs in this model are not due to generic folded singularities, but rather due to singularities at a cusp -- not a fold -- of the critical manifold. Using blowup, we determine the number of SAOs analytically, showing -- as for the folded nodes -- that they are determined by the Weber equation and the ratio of eigenvalues. We also show that the model undergoes a (symmetric) saddle-node bifurcation in the desingularized reduced problem, which -- although resembling a folded saddle-node (type II) at this level -- also occurs on a cusp, and not a fold. We demonstrate that this bifurcation is associated with the emergence of an invariant cylinder, the onset of SAOs, as well as SAOs of increasing amplitude. We relate our findings with numerical computations and find excellent agreement.
