Bifurcation Analysis in a Continuous-Time Information Model with Discrete and Distributed Delays

TL;DR

This paper analyzes a continuous-time social network information model with delays, identifying stability conditions and demonstrating how delays induce Hopf bifurcations, leading to oscillatory behaviors.

Contribution

It introduces a bifurcation analysis framework for a delayed information model, linking delay parameters to stability and oscillation onset.

Findings

Stability conditions for equilibria are established.

Hopf bifurcations occur as delay parameters cross critical thresholds.

The direction of bifurcation depends on delay-related conditions.

Abstract

In this paper, we consider a continuous-time model with discrete and dis-tributed delays to describe how two pieces of information interact in online social networks. Sufficient conditions are carried out to illustrate the stability of each e-quilibrium. Taking time delay as a bifurcation parameter, the system undergoes a sequence of Hopf bifurcation when this parameter passes through a critical value. By methods of multiple scales, we prove that the direction of Hopf bifurcation is depending on the condition which is related to delay.