Mark A. Pinsky
Abstract
This paper introduces a new method for assessing the boundedness and stability of certain vector nonlinear systems with delays and variable coefficients. The approach is based on developing scalar counterparts to the given vector systems. We prove that the solutions to these scalar nonlinear equations, which also include delays and variable coefficients, provide upper bounds for the norms of solutions to the original vector equations if the history functions for both equations are properly matched. This enables the evaluation of the boundedness and stability characteristics of a vector system by analyzing the abridged dynamics of its scalar counterparts. This assessment can be carried out through straightforward simulations or by applying simplified analytical methods. As a result, we introduce new criteria for boundedness and stability and estimate the radii of the balls that contain history functions stemming bounded or stable solutions for the original vector systems. Finally, we validate our inferences through representative simulations that also assess the accuracy of our approach.