Novel adaptive fixed-time synchronization for fractional-order uncertain Hopfield neural networks with proportional delays of unbounded nature

The fixed-time synchronization of fractional-order uncertain Hopfield neural networks (FOUHNNs) with proportional delays is investigated in this paper. Proportional delays, unlike constant or bounded variable delays, are time-varying and unbounded, thereby posing challenges to the synchronization control of fractional master-slave systems. To accomplish fixed-time synchronization, we propose a novel adaptive controller composed of multiple modules, in which the control intensity can be adjusted automatically. It consists of three essential functional modules: one providing negative feedback, one suppressing the adverse influence of proportional delays, and one enhancing the convergence speed. By applying stability lemmas alongside inequality techniques, new synchronization criteria for FOUHNNs are obtained based on the Lyapunov function method. The settling time, which is governed by the control parameters, can be effectively estimated. Numerical experiments are conducted to investigate the influence of control parameters on synchronization settling time.