Let {σ_t}_{t∈(-∞,∞)} be a one-parameter family of hyperbolic Riemannian metrics on an open annulus which is continuouswith respect to the Gromov-Hausdorff topology. We consider a system E_t of ordinary differential equations with singular points which depends on the Riemannian metric σ_t. If t ≠ 0, all of the singular points of E_t are regular. If t = 0, E₀ has an irregular singular point. In this paper, we investigate the behavior of the singular points of E_t . We show that a regular singular point of E_t , together with another regular singular point of E_t , becomes the irregular singular point of E₀ as t (›0) tends to zero and that the irregular singular point of E₀ becomes a non-singular point of E_t as t decreases from zero.