Summary: In this paper, a class of nonlinear BVP (boundary value problem) with bifurcation parameter \(\lambda\) is studied by using singularity theory. We study the nonlinear differential system \(\phi (u,\lambda)=u''+au'+bu+F (u,\lambda)=0\) with conditions: \(u (0)=u (\pi)=0\), where \(a\neq 0\) and the nonlinear term \(F: (\mathbb{R}\times \mathbb{R}, 0)\rightarrow (\mathbb{R},0)\) is a bifurcation problem with finite codimension. The local bifurcation properties of equilibrium solution of the system are given, including the information of the existence and the numbers of the bifurcation solutions.