Projective synchronization of fractional-order uncertain Caputo–Hadamard and quaternion-valued systems with time-varying delay

Abstract In this paper, we investigate two synchronization problems for fractional-order nonlinear systems with uncertainties and time delays. First, the master–slave synchronization of Caputo–Hadamard fractional-order nonlinear systems with uncertainties and time delays are investigated in this research. To begin, a corresponding Lyapunov function is constructed for Caputo–Hadamard fractional-order systems. Using the fractional-order Halanay inequality and the properties of Caputo–Hadamard fractional derivatives, sufficient conditions are obtained to achieve synchronization of uncertain time-delayed nonlinear systems in terms of linear matrix inequalities. The proposed method is tested and verified through two numerical examples. Second, the projective synchronization of fractional-order quaternion-valued inertial neural networks (FQVINNs) with time-varying delays and parameter uncertainties are explores. By applying an appropriate variable transformation, the considered system is converted into a standard fractional-order framework, enabling more tractable theoretical analysis. A unified set of synchronization conditions is then established by designing a quaternion-valued feedback controller. To guarantee projective synchronization, sufficient conditions are derived using fractional-order differential inequalities together with a carefully constructed Lyapunov functional. Finally, numerical simulations are conducted to demonstrate the validity and effectiveness of the proposed both synchronization methods.