Ozone, an important gaseous biosignature, is not only critical for surface UV habitability, but also a proxy for the detection of oxygen, another important gas related to photosynthesis. Detecting ozone on exoplanets is important for assessing their habitability. This study focuses on tidally-locked Earth-like planets around M dwarfs, which outnumber all other stellar types in the Galaxy. Planets in the habitable zone around M dwarfs tend to be tidally locked due to their short orbital periods, resulting in unusual circulation patterns. Recent GCM simulations suggest that around UV-active M dwarfs, Earth-like planets can develop ozone layers with abundance close to that of Earth. Two ozone concentration maxima could form on the nightside, especially within two high-latitude gigantic cyclones. Therefore, these two ozone concentration maxima can serve as a proxy for nightside cyclones detection, which can further validate the GCM-simulated atmospheric circulation. Moreover, it is also an approach of detecting atmospheric circulations on Earth-sized exoplanets. Studying the detectability of such a spatial distribution can provide guidance for future missions on wavelength selection and observing strategy. Given the fact that the capability of current telescopes and those in the foreseeable future would not be able to resolve exoplanets better than one pixel, our analysis focuses on disk-integrated single-point light curves. Reflected and emitted light of the simulated Earth-like exoplanet are generated using a radiative transfer model and then integrated over the planetary disk to create a single-point source, which follows with time series analysis. Detectability of ozone distribution and climate system are derived from the synthetic light curves combined with viewing geometry.