This paper is concerned with the globally asymptotic stability of the Riemann‐Liouville fractional‐order neural networks with time‐varying delays. The Lyapunov functional approach to stability analysis for nonlinear fractional‐order functional differential equations is discussed. By constructing an appropriate Lyapunov functional associated with the Riemann‐Liouville fractional integral and derivative, the asymptotic stability criteria of fractional‐order neural networks with time‐varying delays and constant delays are derived. The advantage of our proposed method is that one may directly calculate the first‐order derivative of the Lyapunov functional. Two numerical examples are also presented to illustrate the validity and feasibility of the theoretical results. With the increasing of the order of fractional derivatives, the state trajectories of neural networks show that the speeds of converging toward zero solution are faster and faster.

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This paper investigates the existence and globally asymptotic stability of equilibrium solution for Riemann-Liouville fractionalorder hybrid BAM neural networks with distributed delays and impulses. The factors of such network systems including the distributed delays, impulsive effects, and two different fractional-order derivatives between the -layer and -layer are taken into account synchronously. Based on the contraction mapping principle, the sufficient conditions are derived to ensure the existence and uniqueness of the equilibrium solution for such network systems. By constructing a novel Lyapunov functional composed of fractional integral and definite integral terms, the globally asymptotic stability criteria of the equilibrium solution are obtained, which are dependent on the order of fractional derivative and network parameters. The advantage of our constructed method is that one may directly calculate integer-order derivative of the Lyapunov functional. A numerical example is also presented to show the validity and feasibility of the theoretical results.

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Owing to the symmetry between drive–response systems, the discussions of synchronization performance are greatly significant while exploring the dynamics of neural network systems. This paper investigates the quasi-synchronization (QS) and quasi-uniform synchronization (QUS) issues between the drive–response systems on fractional-order variable-parameter neural networks (VPNNs) including probabilistic time-varying delays. The effects of system parameters, probability distributions and the order on QS and QUS are considered. By applying the Lyapunov–Krasovskii functional approach, Hölder’s inequality and Jensen’s inequality, the synchronization criteria of fractional-order VPNNs under controller designs with constant gain coefficients and time-varying gain coefficients are derived. The obtained criteria are related to the probability distributions and the order of the Caputo derivative, which can greatly avoid the situation in which the upper bound of an interval with time delay is too large yet the probability of occurrence is very small, and information such as the size of time delay and probability of occurrence is fully considered. Finally, two examples are presented to further confirm the effectiveness of the algebraic criteria under different probability distributions.