Opinions dynamics for a generalized nonlinear model with state-dependent susceptibility in signed networks on time scales

Abstract

This article studies the opinions dynamics for a generalized nonlinear model with state-dependent susceptibility in signed networks on time scales. Unlike continuous-time or discrete-time models, time-scale systems can integrate continuous and discrete systems within a unified framework. Nevertheless, time-scale induced discontinuity properties can make it difficult to study the well-posedness and dynamics analysis of the opinions dynamics. In this paper, we first give a time-scale related condition to ensure the well-posedness of the nonlinear model. Then, based on time-scale theory, the comparison principle and the generalized Gronwall inequality, we focus on analyzing the dynamic behavior of the nonlinear model with three distinct susceptibility functions, which represent the behavior pattern of stubborn positives, stubborn extremists, and stubborn neutrals respectively. For the behavior pattern of stubborn positives and stubborn extremists, some sufficient conditions are given such that all agents’ viewpoints will converge exponentially towards a specific equilibrium point or achieve consensus under certain initial conditions. For the behavior pattern of stubborn neutrals, all agents’ opinions will finally reach neutrality.