We investigate the Ambrosetti-Prodi periodic problem for a Minkowski-curvature differential equation with an indefinite attractive singularity \(\begin{aligned} \left( \frac{x'}{\sqrt{1 - x'^2}} \right) '+f(x)x'+a(t)x^{\delta }+\frac{b(t)}{x^\rho }=s, \end{aligned}\) where \(f \in C((0, +\infty ); \mathbb {R})\) , \( a , b \in C(\mathbb {R}/T\mathbb {Z}; \mathbb {R}) \) are T-periodic functions satisfying \(b(t)\ge 0\) . Several Ambrosetti-Prodi type results on the existence and multiplicity of periodic solutions are obtained under different conditions. We apply the main results to a classical MEMS model within a relativistic regime, which is closely connected with an open problem raised by P. J. Torres on saddle-node bifurcation of periodic solution. Through numerical bifurcation analysis, we verify the main theorems and further explore the bifurcations of periodic solutions when the model admits both small and large period T. It is revealed that, when T is small, the resulting two periodic solutions exhibit a classical stability pattern, consisting of one asymptotically stable and one unstable solution. For larger periods, this pattern breaks down as the model develops complex dynamical behavior, such as period-doubling and multiple saddle-node bifurcations of periodic solutions. The interplay of these bifurcations leads to multiple periodic solutions, multistable periodic solutions, and higher-order periodic solutions.
