Mittag‐Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field

Fractional order quaternion‐valued neural networks are a type of fractional order neural networks for which neuron state, synaptic connection strengths, and neuron activation functions are quaternion. This paper is dealing with the Mittag‐Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field. The fractional order quaternion‐valued neural networks are separated into four real‐valued systems forming an equivalent four real‐valued fractional order neural networks, which decreases the computational complexity by avoiding the noncommutativity of quaternion multiplication. Via some fractional inequality techniques and suitable Lyapunov functional, a brand new criterion is proposed first to ensure the Mittag‐Leffler stability for the addressed neural networks. Besides, the combination of quaternion‐valued adaptive and impulsive control is intended to realize the asymptotically synchronization between two fractional order quaternion‐valued neural networks. Ultimately, two numerical simulations are provided to check the accuracy and validity of our obtained theoretical results.