Homoclinic organization in the Hindmarsh-Rose model: A three parameter study.
Homoclinic organization in the Hindmarsh-Rose model: A three parameter study.
复制标题
Hindmarsh-Rose 模型中的同宿组织：三参数研究。
作者：
R. Barrio;S. Ibáñez;L. Pérez
爆裂现象广泛存在于各种快慢系统中。在这篇文章中，我们考虑Hindmarsh-Rose神经元模型，其中，正如文献中所知的那样，爆发动力学中存在同宿分支。然而，全球同宿结构还远未完全被理解。在一个三参数空间中，我们的数值分析结果显示了一个复杂的分支图谱，它从奇异极限延伸到快慢视角不再适用的区域。基于这些信息，我们提出了一个全局的理论描述。在快慢区，余维一同宿分叉的曲面彼此指数接近。值得注意的是，通过这些表面的特殊性质，我们展示了当适当的二维切片在三参数空间中被考虑时，Hindmarsh-Rose模型是如何显示同宿分支的孤立子的。另一方面，这些同宿分叉曲面包含与参数值相对应的曲线，其中显示了额外的退化。这些余维二分叉曲线组织了与尖峰相加过程相关的分叉，它们的行为就像“书脊”，聚集了周期轨道分叉的“页面”。根据参数空间的探索方式，同宿现象可能不存在，也可能远离，但它们在爆发动力学中的组织作用是毋庸置疑的，因为涉及到的分叉是在这些现象中产生的。这一点在全局分析和建议的理论方案中得到了证明。
Bursting phenomena are found in a wide variety of fast-slow systems. In this article, we consider the Hindmarsh-Rose neuron model, where, as it is known in the literature, there are homoclinic bifurcations involved in the bursting dynamics. However, the global homoclinic structure is far from being fully understood. Working in a three-parameter space, the results of our numerical analysis show a complex atlas of bifurcations, which extends from the singular limit to regions where a fast-slow perspective no longer applies. Based on this information, we propose a global theoretical description. Surfaces of codimension-one homoclinic bifurcations are exponentially close to each other in the fast-slow regime. Remarkably, explained by the specific properties of these surfaces, we show how the Hindmarsh-Rose model exhibits isolas of homoclinic bifurcations when appropriate two-dimensional slices are considered in the three-parameter space. On the other hand, these homoclinic bifurcation surfaces contain curves corresponding to parameter values where additional degeneracies are exhibited. These codimension-two bifurcation curves organize the bifurcations associated with the spike-adding process and they behave like the "spines-of-a-book," gathering "pages" of bifurcations of periodic orbits. Depending on how the parameter space is explored, homoclinic phenomena may be absent or far away, but their organizing role in the bursting dynamics is beyond doubt, since the involved bifurcations are generated in them. This is shown in the global analysis and in the proposed theoretical scheme.
影响因子：
2
作者：
Tsaneva-Atanasova K;Osinga HM;Riess T;Sherman A
通讯作者：
Sherman A
瞬态突发中尖峰添加机制的动力学系统分析。
影响因子：
2.3
作者：
Nowacki J;Osinga HM;Tsaneva-Atanasova K
通讯作者：
Tsaneva-Atanasova K