The atomic entanglement dynamics is studied in the double Jaynes-Cummings model composed of two coupled cavities. Effect of the cavity-cavity coupling constant A and the atomic initial states on entanglement sudden death (ESD) and entanglement sudden birth (ESB) is analysed. By studying the atomic concurrence, it was shown that when the atomic initial states are the first kind Bell-like states, ESD and ESB doesn’t appear, coupling of the cavities can increase the atomic entanglement and entanglement occurs even the initial states are disentangled. When the atomic initial states are the second kind Bell-like states, the time intervals between the ESD and ESB, and whether the ESD occurs or not are controlled by the coupling constant A. The atomic entanglement can’t be produced by the coupling between the cavities when the atomic initial states are disentangled and in the second kind Bell-like states.