We consider smooth systems limiting as $ϵ\to 0$ to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with sufficiently small but non-zero ϵ, using a combination of geometric singular perturbation theory and blow-up. We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an ϵ−dependent domain which shrinks to zero as $ϵ\to 0$, identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation oscillations in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation oscillations to regular cycles within the ϵ−dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction.

我们认为光滑系统限制为 $\epsilon \to 0$到具有边界聚焦 (BF) 分叉的分段平滑 (PWS) 系统。在推导出合适的局部范式后，我们使用几何奇异扰动理论和爆炸的组合研究具有足够小但非零ϵ的光滑系统的动力学。我们展示了 PWS 系统中 BF 分叉的类型决定了平滑系统在ϵ -依赖域内的分叉结构，该域收缩为零$\epsilon \to 0$，在一种情况下确定超临界 Andronov-Hopf 分岔，在另外两种情况下确定超临界 Bogdanov-Takens 分岔。我们还表明，与 BF 分叉相关的 PWS 循环在光滑系统中作为弛豫振荡持续存在，并证明存在一系列稳定的极限循环，将弛豫振荡与上述依赖于ϵ 的域内的规则循环连接起来。我们的结果应用于 Gause 捕食者-猎物相互作用和受摩擦影响的机械振荡模型。