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 multidimensional case the results obtained in a previous paper where the slow-time system is 1-dimensional. We prove the existence of a unique trajectory (x(t,

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... λ),y(t, ε, λ)) 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 (x(t, ε, λ),y(t, ε, λ)) is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples.