Some Dynamical Behaviors of Fractional-Order Commutative Quaternion-Valued Neural Networks via Direct Method of Lyapunov

Some dynamical behaviors of fractional-order commutative quaternion-valued neural networks (FCQVNNs) are studied in this paper. First, because the commutative quaternion does not satisfy Schwartz triangle inequality, the FCQVNNs are divided into four real-valued neural networks (RVNNs) through quaternion commutative multiplication rules. Furthermore, several types of dynamical behaviors including global Mittag-Leffler stability, the boundedness with bounded disturbances, complete synchronization and quasi-synchronization of FCQVNNs are studied. Simultaneously, several conditions for these dynamical behaviors are driven by fractional-order Lyapunov direct method, some inequality techniques and fractional differential equation theory. At last, the effectiveness and feasibility of the obtained theoretical results are verified by several numerical simulation examples.