We consider a singularly perturbed system where the fast dynamics of the unperturbed problem exhibits a trajectory homoclinic to a critical point. We assume that the slow time system admits a unique critical point, which undergoes a bifurcation as a second parameter varies: transcritical, saddle-node, or pitchfork. We generalize to the multi-dimensional case the results obtained in a previous paper where the slow-time system is $1$-dimensional. We prove the existence of a unique trajectory $(reve{x}(t,arepsilon,lambda),reve{y}(t,arepsilon,lambda))$ homoclinic to a centre manifold of the slow manifold. Then we construct curves in the $2$-dimensional parameters space, dividing it in different areas where $(reve{x}(t,arepsilon,lambda),reve{y}(t,arepsilon,lambda))$ is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples.
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机译：我们考虑一个奇异摄动系统，其中未摄动问题的快速动力学表现出到临界点的同质轨迹。我们假设慢速系统承认一个独特的临界点，该临界点会随着第二个参数的变化而分叉：跨临界，鞍形节点或干草叉。我们将多维情况推广到以前的论文中，其中慢速系统为1维维。我们证明慢速流形的中心流形的唯一轨迹$（ breve {x}（t， varepsilon， lambda）， breve {y}（t， varepsilon， lambda））$存在。然后我们在$ 2 $维参数空间中构造曲线，将其划分为$（ breve {x}（t， varepsilon， lambda）， breve {y}（t， varepsilon， lambda） ）$是同宿，异宿或无界的。我们为这些曲线的切线得出明确的公式。通过一些示例说明了结果。
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