Modeling the Bifurcation Dynamics of Rumor Propagation in the Spatial Environment

Linhe Zhu, Xinlin Chen
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Abstract

The harm caused by rumors is immeasurable. Studying the dynamic characteristics of rumors can help control their spread. In this paper, we propose a nonsmooth rumor model with a nonlinear propagation rate. First, we utilize the positive invariant regions to prove the boundedness of solutions. Second, we analyze the conditions for the existence of equilibrium points in both the left and right systems. Additionally, we confirm the occurrence of saddle-node bifurcation in the left system. Next, by considering the influence of spatial diffusion, we establish the conditions for Turing instability. Then we discuss the conditions for spatial homogeneous and inhomogeneous Hopf bifurcations in the left and right systems, respectively. We differentiate between supercritical and subcritical bifurcations using the Lyapunov coefficient. Furthermore, we examine the conditions for the existence of discontinuous Hopf bifurcation at the demarcation point. Finally, in the numerical simulation section, we validate our theorems on Turing patterns. We also investigate the impact of parameter changes on rumor propagation and conclude that an increase in the psychological inhibitory factor significantly reduces the rate of rumor propagation, providing an effective strategy for curbing rumors. To that end, we fit actual data to our system and the results are excellent, confirming the validity of the system.
空间环境中谣言传播的分岔动态建模
谣言造成的危害不可估量。研究谣言的动态特征有助于控制谣言的传播。本文提出了一个具有非线性传播率的非光滑谣言模型。首先,我们利用正不变区域来证明解的有界性。其次,我们分析了左右系统中平衡点存在的条件。此外,我们还证实了左系统中鞍节点分岔的发生。接下来,通过考虑空间扩散的影响,我们确定了图灵不稳定性的条件。然后,我们分别讨论了左系统和右系统出现空间均质和非均质霍普夫分岔的条件。我们用 Lyapunov 系数来区分超临界和亚临界分岔。此外,我们还研究了在分界点存在不连续霍普夫分岔的条件。最后,在数值模拟部分,我们在图灵模式上验证了我们的定理。我们还研究了参数变化对谣言传播的影响,并得出结论:心理抑制因子的增加会显著降低谣言的传播速度,从而为遏制谣言提供有效策略。为此,我们将实际数据拟合到我们的系统中,结果非常出色,证实了系统的有效性。
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