A New Approach to Reducing Vector Delay Nonlinear Systems: Boundedness and Stability Analysis

TL;DR

This paper presents a novel scalar-based method for evaluating the boundedness and stability of vector nonlinear delay systems, simplifying analysis and providing new criteria validated through simulations.

Contribution

It introduces a scalar counterpart approach to assess stability and boundedness in vector systems with delays, offering new analytical criteria and validation methods.

Findings

Scalar equations provide upper bounds for vector solutions

New criteria for boundedness and stability are established

Simulations confirm the effectiveness and accuracy of the approach

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.