Barrio Roberto, Ibáñez Santiago, Pérez Lucía
Departamento de Matemática Aplicada and IUMA, Computational Dynamics group, University of Zaragoza, 50009, Zaragoza, Spain.
Departamento de Matemáticas, University of Oviedo, 33007, Oviedo, Spain.
Sci Rep. 2024 May 13;14(1):10900. doi: 10.1038/s41598-024-61574-6.
By studying a nonlinear model by inspecting a p-dimensional parameter space through -dimensional cuts, one can detect changes that are only determined by the geometry of the manifolds that make up the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, in particular, of the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are illustrated in two models of neuron activity: the Hindmarsh-Rose and the FitzHugh-Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter, but they should be of interest to anyone studying bifurcation diagrams in the context of nonlinear phenomena.
通过研究一个非线性模型，通过 - 维切割来检查一个p维参数空间，人们可以检测到仅由构成分岔集的流形的几何形状所决定的变化。我们将这些变化称为几何分岔。它们可以在可微映射的奇点理论框架内，特别是在莫尔斯理论的框架内得到理解。在一个三维参数空间中工作，几何分岔在两个神经元活动模型中得到了说明： Hindmarsh-Rose模型和FitzHugh-Nagumo系统。两者都是具有一个控制慢变量时间尺度的小参数的快慢系统。在对应于这个特殊小参数固定值的切片上观察到几何分岔，但它们应该会引起任何在非线性现象背景下研究分岔图的人的兴趣。
