Stability Analysis of Axion Dark Energy Model

We investigate the stability of an axion dark energy model characterized by a periodic scalar-field potential motivated by pseudo Nambu Goldstone bosons. While such models are well suited to drive late-time cosmic acceleration and can exhibit rich dynamical features, including oscillatory behavior of the dark-energy equation-of-state parameter, their viability crucially depends on the existence of stable cosmological solutions. By reformulating the background evolution equations as an autonomous dynamical system, we identify the fixed points corresponding to radiation-dominated, matter-dominated, and dark energy dominated epochs. Using both analytical and numerical techniques, we analyze the stability properties of these fixed points within the observationally allowed region of parameter space. Our results show that the radiation- and matter-dominated phases emerge naturally as saddle points, ensuring a standard cosmological evolution, while the late-time dark energy dominated solution acts as a stable attractor. This guarantees the robustness of cosmic acceleration against variations in initial conditions. Overall, our study establishes the axion dark energy model as a dynamically consistent and observationally viable candidate for explaining the late-time accelerated expansion of the Universe.