The antenna is subjected to the complicated space environment, which results in nonlinear large amplitude vibration. These nonlinear vibrations severely affect the stability and performance of the structure. Due to the energy exchange between different modes, the circular mesh antenna is easier to produce complex internal resonance. The internal resonance plays a dominant role in large amplitude nonlinear vibrations of the circular mesh antenna. The circular mesh antenna is simplified as the equivalent circular cylindrical shell in this paper. According to the finite element analysis, it is possible that the 1 ∶1 internal resonance between the fourth order mode and the fifth order mode of the circular mesh antenna occurs. Therefore, the local dynamics of the structure with 1 ∶1 internal resonances are investigated by using theoretical analysis and numerical simulations. In this paper, the stability of the equilibrium point under small perturbations is studied. Two types of critical equilibrium points are considered, such as a double zero and two negative eigenvalues, as well as a double zero and a pair of purely imaginary eigenvalues. The stable regions and the unstable regions of the initial equilibrium point and the critical bifurcation curves are obtained by the center manifold theorem, nonlinear?transform and Routh?Hurwitz criterion. Base on the averaged equation of the circular mesh antenna, the trajectories of the initial equilibrium point are examined in order to verify theoretical results.