SIAM Journal on Mathematical Analysis, Volume 53, Issue 4, Page 3717-3758, January 2021.
For a slow-fast system of the form $\dot{p}=\epsilon f(p,z,\epsilon)+h(p,z,\epsilon)$, $\dot{z}=g(p,z,\epsilon)$ for $(p,z)\in \mathbb R^n\times \mathbb R^m$, we consider the scenario that the system has invariant sets $M_i=\{(p,z): z=z_i\}$, $1\le i\le N$, linked by a singular closed orbit formed by trajectories of the limiting slow and fast systems. Assuming that the stability of $M_i$ changes along the slow trajectories at certain turning points, we derive criteria for the existence and stability of relaxation oscillations for the slow-fast system. Our approach is based on a generalization of the entry-exit relation to systems with multi-dimensional fast variables. We then apply our criteria to several predator-prey systems with rapid ecological evolutionary dynamics to show the existence of relaxation oscillations in these models.

中文翻译：

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多维慢-快系统中的弛豫振荡和进入-退出函数

SIAM 数学分析杂志，第 53 卷，第 4 期，第 3717-3758 页，2021 年 1 月。
对于形式为 $\dot{p}=\epsilon f(p,z,\epsilon)+h(p,z,\epsilon)$ 的慢速系统，$\dot{z}=g(p, z,\epsilon)$ 为 $(p,z)\in \mathbb R^n\times \mathbb R^m$，我们考虑系统具有不变集 $M_i=\{(p,z) 的场景： z=z_i\}$, $1\le i\le N$，由有限慢速和快速系统的轨迹形成的奇异闭合轨道连接。假设$M_i$ 的稳定性在某些转折点沿慢轨迹变化，我们推导出慢-快系统松弛振荡的存在和稳定性的标准。我们的方法基于对具有多维快速变量的系统的进入-退出关系的概括。然后，我们将我们的标准应用于具有快速生态进化动力学的几个捕食者-猎物系统，以显示这些模型中存在弛豫振荡。