Fractional discrete neural networks with variable order: solvability, finite time stability and synchronization

Research in the field of dynamic behaviors in neural networks with variable-order differences is currently a thriving area, marked by various significant discoveries. However, when it comes to discrete-time neural networks featuring fractional variable-order nonlocal and nonsingular kernels, there has been limited exploration. This paper stands as one of the initial contributions to this subject, focusing primarily on the topics of stability and synchronization in finite-time within discrete neural networks. The research employs the nabla ABC variable-order difference operator, with a primary approach involving the investigation of a novel Gronwall inequality using the Atangana-Baleanu difference variable-order sum operator. This analysis leads to the development of a uniqueness theorem and a criterion for the stability in finite-time of variable-order discrete neural networks. Furthermore, the requirements stemming from this type of stability and the novel Gronwall inequality serve as the foundation for establishing the conditions necessary for achieving finite-time synchronization in these networks, employing a specific control using state feedback method. Finally, the study utilizes numerical solutions to validate the obtained results.