论文标题
三个时期系统中混合模式振荡的分叉：扩展的原型示例
Bifurcations of mixed-mode oscillations in three-timescale systems: an extended prototypical example
论文作者
论文摘要
我们研究一类普通微分方程的多参数的三维系统，这些系统在三个不同的时间尺度上表现出动态。我们应用几何奇异扰动理论来探索这些系统对其参数的几何形状的依赖性，重点是混合模式振荡（MMOS）及其分叉。特别是，我们发现了一种新颖的几何机制，该机制编码了从具有小振幅振荡（SAO）的单个时期的MMO向具有双重位置SAOS的MMO。我们确定了一个相对简单的原型三频率系统，该系统实现了我们的机制，具有一维$ s $形的超临界流形，以对称方式嵌入二维$ s $ S $形状的临界流形中。我们表明，化学动力学的KOPER模型仅是该原型系统的特定认识，用于特定的参数选择。特别是，我们解释了与其中的双重时期的混合模式动力学的鲁棒性发生。最后，我们认为我们的几何机制可以阐明具有相似基础几何形状的更复杂系统的混合模式动力学，例如从数学神经科学中降低了Hodgkin-Huxley方程的三维，三维时数。
We study a class of multi-parameter three-dimensional systems of ordinary differential equations that exhibit dynamics on three distinct timescales. We apply geometric singular perturbation theory to explore the dependence of the geometry of these systems on their parameters, with a focus on mixed-mode oscillations (MMOs) and their bifurcations. In particular, we uncover a novel geometric mechanism that encodes the transition from MMOs with single epochs of small-amplitude oscillations (SAOs) to those with double-epoch SAOs. We identify a relatively simple prototypical three-timescale system that realises our mechanism, featuring a one-dimensional $S$-shaped supercritical manifold that is embedded into a two-dimensional $S$-shaped critical manifold in a symmetric fashion. We show that the Koper model from chemical kinetics is merely a particular realisation of that prototypical system for a specific choice of parameters; in particular, we explain the robust occurrence of mixed-mode dynamics with double epochs of SAOs therein. Finally, we argue that our geometric mechanism can elucidate the mixed-mode dynamics of more complicated systems with a similar underlying geometry, such as of a three-dimensional, three-timescale reduction of the Hodgkin-Huxley equations from mathematical neuroscience.
