Estimating the Evolution of Solution Norms in Vector Delay Nonlinear Systems: Stability and Boundedness

Existing methods rarely capture the temporal evolution of solution norms in vector nonlinear DDEs with variable delays and coefficients, often leading to overly conservative boundedness and stability criteria. We develop a framework that constructs scalar counterparts of vector DDEs whose solutions upper-bound the evolution of the original solution norms when the corresponding history functions are matched. This reduction enables boundedness and stability assessment of vector DDEs through the dynamics of their scalar counterparts, using straightforward simulations or simplified analytical reasoning. New boundedness and stability criteria and the estimates of the radii of balls containing history functions that yield bounded or stable solutions for the original vector systems were derived and validated through representative simulations.