Quasi-Synchronization and Quasi-Uniform Synchronization of Caputo Fractional Variable-Parameter Neural Networks with Probabilistic Time-Varying Delays

Abstract

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.

Event‐triggered delayed impulsive control for quasi‐synchronization of complex dynamic systems with packet loss and parameter mismatch

“…Case(b): Suppose {t k } is generated by the event trigger function ‖Ξ(t )‖ > 0 in Equation (7). The Dini derivative of ‖Θ(t )‖ with respect to t along the solution of Equation ( 8) is obtained by…”

“…Suppose that (H1), (H2) holds. Under the controller (5) with Equation (7), if there exist some positive constants 𝜎 1 , 𝜎 2 , 𝜎 3 , 𝜌, 𝜌 1 , 𝜌 2 , 𝜌 3 , 𝜇 such that…”

“…Specifically, if ‖Ξ(t )‖ > 0 and the mechanism is not triggered, we update t k+1 to t k + Δ. This ensures that the inefficiency of event mechanisms (7) or ( 25) is avoided [16,26].…”

“…Case 1: (Event-triggered control strategy) From Equation ( 4), it has ‖ (t , x(t ))‖ = ‖ x(t − 0.15)(1 − x(t − 0.15))‖ ≤ 0.99 = 𝜍. In Equations ( 5) and (7), take K = −94, L = 0.1, 𝜏 k = 0.005. △ = 0.1, 𝜁 = 0.01.…”

“…However, two or more system synchronization error limits must be within an acceptable range. In this context, quasi-synchronization has been proposed to consider this particular synchronization phenomenon, and it has gradually revealed its potential applications in various fields, and has achieved many excellent results [3,6,7]. However, in the case of large parameter mismatch, it will reduce the quality of synchronization, and even lead to failure to achieve synchronization.…”