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This paper deals with the problems of quasi-projective synchronization (QPS) and finite-time synchronization (FTS) for a kind of delayed fractional-order BAM neural networks (DFOBAMNNs). In order to reach the goals of synchronization and more accurately gauge of settling time and error level, several fresh quantized controllers are structured to make the utmost of confined communication resources. Then, based on the finite-time theorem, quantized control strategy, Lyapunov function theory and properties of Mittag-Leffler function as well as inequality analysis techniques, some plentiful criteria are formed to set up a relation between control gains and quantization parameters. In addition, the corresponding error bound of QPS and guages of the settling time on FTS are also given. Finally, a few numerical examples are introduced to validate the effectiveness of the presented control protocols.

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