A honeycomb sandwich plate with hexagonal honeycomb core was investigated to reveal the dynamic behavior near a critical point characterized by initial resonance. Based on the averaged equations, the transition boundaries were obtained to divide the parameter space into a set of regions, which correspond to different types of solutions. By applying the stability criteria to determine the stable conditions of respective equilibrium points, the conditions of the occurrence of double Hopf bifurcations were found. Two types of periodic solutions may bifurcate from the initial equilibrium. And the periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori.