One type of mode-hopping detection system based on fiber Michelson interferometer is set up. The influence of pump modulation on the mode-hopping characteristics of ultra-narrow linewidth erbium-doped fiber ring laser is investigated experimentally through the system. The mode-hopping map of laser labeled by pump current is measured to reveal the mode-hopping characteristics. Numerical results show that the laser will orderly experience steady single-mode area, unsteady single-mode area and multimode area with the pump current increasing, and there exists a second threshold over which the laser will hop from single-mode into drastic multimode. Besides, frequency drift going with serial mode hops can be excited by pump-induced thermal effect or hostile disturbances applied on the laser cavity.