Dynamics and synchronization of a fractional conformable neural network with power-law

In this work, we studied the dynamics of a chaotic neural network using the fractional-conformable operator and non-integer order derivatives. We first systematically show the dynamics behavior of the network using the conformable operator. Then, we show the dynamics of the network using fractional-order derivatives. We compared the different effects and behaviors when using conformable operators and fractional-order derivatives. Next, we studied the dynamics when applying both operators to the network simultaneously, and we graphically show the effects of each operator on the chaotic neural network. Following the dynamical analysis, we used a fractional-variable order in the derivative operator to compare the system behavior when we used a fractional constant order in the derivative operator. We used a smooth function in all our simulations in the interval (0, 1]. Our results show that either the conformable operator or the fractional derivative, both of them, affect the dynamics of the network showing interesting behaviors. We show that when the order in the conformable operator decreases an attractor is generated which takes the system into a stable state, and when the order in the fractional derivative decreases the system loses its dynamics. Contrary, when we used the fractional-variable order in the derivative operator, the system does not lose its dynamics; instead, it gains more chaotic behavior, which means that the system is susceptible to the initial conditions fractional-variable order function and its initial point. Finally, after depicting the different system behavior under both operators, we proposed a full master-slave synchronization using the fractional-conformable approach. We designed a fractional order controller to reach synchronization using optimal gains means Cuckoo search algorithm. Our results show the different behaviors when using the fractional-conformable approach in the master and slave systems. As we expected, the gains in a combination of the non-integer order allow achieving the synchronization between the master and the slave system even when the dynamics in the slave system was regular.