Title：Homoclinic Orbits of the FitzHugh-Nagumo Equation Bifurcations in the full system

Abstract：This paper investigates travelling wave solutions of the FitzHugh-Nagumo equation from the viewpoint of fast-slow dynamical systems. These solutions are homoclinic orbits of a three dimensional vector field depending upon system parameters of the FitzHugh-Nagumo model and the wave speed. Champneys et al. [A.R. Champneys, V. Kirk, E. Knobloch, B.E. Oldeman, and J. Sneyd, When Shilnikov meets Hopf in excitable systems, SIAM Journal of Applied Dynamical Systems, 6(4), 2007] observed sharp turns in the curves of homoclinic bifurcations in a two dimensional parameter space. This paper demonstrates numerically that these turns are located close to the intersection of two curves in the parameter space that locate non-transversal intersections of invariant manifolds of the three dimensional vector field. The relevant invariant manifolds in phase space are visualized. A geometrical model inspired by the numerical studies displays the sharp turns of the homoclinic bifurcations curves and yields quantitative predictions about multi-pulse and homoclinic orbits and periodic orbits that have not been resolved
in the FitzHugh-Nagumo model. Further observations address the existence of canard explosions and mixed-mode oscillations.

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标题：（译）Fitzugh-Nagumo方程的同宿轨线：全系统的分岔

摘要：本文从快-慢动力系统的角度研究了Fitzugh-Nagumo方程的行波解。这些解是三维向量场的同宿轨线，该向量场取决于Fitzugh-Nagumo模型的系统参数和波速。Champneys等人[A.R.Champneys，V.Kirk，E.Knobloch，B.E.Oldeman，J.Sneyd，《在可激发系统中同时发现Hopf和Shilnikov时》，SIAM JoUrnal of Applied Dynamical Systems, 6(4), 2007]在二维参数空间中观察到同宿分岔曲线的急剧转弯。本文用数值方法证明了这些转弯位于参数空间中两条曲线的交点附近，该参数空间建立于三维向量场不变流形的非横向交点处。对相空间中的相关不变流形进行了可视化。受数值研究启发的几何模型显示了同宿分岔曲线的急转弯，并给出了Fitzugh-Nagumo模型中尚未解决的多脉冲同宿轨线和周期轨线的定量预测。进一步的观察解决了鸭解式迸发和混合模式振荡的存在。
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