Stability Analysis and Synchronization of Distributed-Order Fractional Nonlinear Dynamical Systems with Time Delay

This paper investigates stability and complete synchronization of distributed-order fractional nonlinear dynamical systems (DONDSs) with a constant time delay. For the continuous Caputo distributed-order model, a Laplace-transform resolvent approach gives the sufficient incremental small-gain condition qw=(L+M)∥Kw∥L1<1 under stated well-posedness and resolvent assumptions. The resolvent requirements are verified for scalar damping A=aI, for which the criterion reduces to L+M<a, and for symmetric positive-definite A, for which L+M<λmin(A) is sufficient; well-posedness remains a separate hypothesis. For the finite multi-term model, the corresponding condition is q=(L+M)∥Km∥L1<1. A conditional history-space generalized Wright Lyapunov criterion is also established, with decay described through a scalar multi-term relaxation function and its generalized Wright series representation subject to additional inversion conditions. We develop an Adams product integration predictor–corrector method, examine contraction of its endpoint corrector, and compare it with backward-Euler convolution quadrature and a fully implicit Adams endpoint. Finite multi-term Liu and financial systems illustrate the framework, accompanied by numerical studies of order distributions, delays, control gains, perturbations, and parameter interactions. Finally, an active control construction cancels nonlinear and delayed mismatch terms under ideal full state and exact model assumptions, leaving a stable linear distributed-order synchronization error system.