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Successive lag synchronization (SLS) for nonlinear dynamical networks is investigated by using aperiodically intermittent control. Different from previous works about the SLS, the proposed controllers could be discontinuous and aperiodic. Aperiodically intermittent controllers are proposed to realize SLS on the dynamical networks with and without communication delay. Furthermore, several sufficient conditions are obtained by applying the Lyapunov function method to make SLS achieve global stability. Finally, a loop-shaped network example and a chain-shaped network example are provided to verify correctness of our results.