This paper examines a spike-adding bifurcation phenomenon whereby small-amplitude canard cycles transition into large-amplitude bursting oscillations along a single continuous branch in parameter space. We consider a class of three-dimensional singularly perturbed ODEs with two fast variables and one slow variable and singular perturbation parameter \(\varepsilon \ll 1 \) under general assumptions which guarantee such a transition occurs. The primary ingredients include a cubic critical manifold and a saddle homoclinic bifurcation within the associated layer problem. The continuous transition from canard cycles to N-spike bursting oscillations up to \(N\sim {\mathcal {O}}(1/\varepsilon )\) spikes occurs upon varying a single bifurcation parameter on an exponentially thin interval. We construct this transition rigorously using geometric singular perturbation theory; critical to understanding this transition are the existence of canard orbits and slow passage through the saddle homoclinic bifurcation, which are analyzed in detail.

本文研究了添加尖峰的分叉现象，由此小幅度的Canard循环沿着参数空间中的单个连续分支转换为大幅度的突发振荡。在一般假设下，我们考虑一类具有两个快速变量和一个慢变量以及奇异摄动参数\（\ varepsilon \ ll 1 \）的三维奇异摄动ODE，它们保证发生这种过渡。主要成分包括立方临界歧管和相关层问题内的鞍形均斜分叉。从Canard循环到N峰值爆发振荡的连续过渡，直至\（N \ sim {\ mathcal {O}}（1 / \ varepsilon）\）在一个指数间隔上改变单个分叉参数时，就会出现尖峰。我们使用几何奇异摄动理论严格构建了这种过渡。理解这一过渡的关键是鸭口轨道的存在和缓慢通过鞍座同斜叉分叉的过程，对此进行了详细分析。