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Bifurcation analysis of an opinion dynamics model coupled with an environmental dynamics

Anthony Couthures, Anastasia Bizyaeva, Vineeth S. Varma, Alessio Franci, Irinel-Constantin Morarescu

TL;DR

The work studies a continuous-time model in which opinion dynamics are coupled to environmental feedback. By exploiting forward invariance of the synchronization manifold, the authors reduce the analysis to the fully synchronized subsystem (FSOE) and perform a rigorous bifurcation analysis around the origin. They derive conditions for singularities of the Jacobian and establish pitchfork and Hopf bifurcations, supported by numerical simulations that demonstrate bistable and oscillatory regimes. The results illuminate how trust in environmental signals, captured by the parameter $\beta$, governs collective behavior and potential environmental oscillations, offering insight into policy-relevant dynamics of climate-related opinions.

Abstract

We consider an opinion dynamics model coupled with an environmental dynamics. Based on a forward invariance argument, we can simplify the analysis of the asymptotic behavior to the case when all the opinions in the social network are synchronized. Our goal is to emphasize the role of the trust given to the environmental signal in the asymptotic behavior of the opinion dynamics and implicitly of the coupled system. To do that, we conduct a bifurcation analysis of the system around the origin when the trust parameter is varying. Specific conditions are presented for both pitchfork and Hopf bifurcation. Numerical illustration completes the theoretical findings.

Bifurcation analysis of an opinion dynamics model coupled with an environmental dynamics

TL;DR

The work studies a continuous-time model in which opinion dynamics are coupled to environmental feedback. By exploiting forward invariance of the synchronization manifold, the authors reduce the analysis to the fully synchronized subsystem (FSOE) and perform a rigorous bifurcation analysis around the origin. They derive conditions for singularities of the Jacobian and establish pitchfork and Hopf bifurcations, supported by numerical simulations that demonstrate bistable and oscillatory regimes. The results illuminate how trust in environmental signals, captured by the parameter , governs collective behavior and potential environmental oscillations, offering insight into policy-relevant dynamics of climate-related opinions.

Abstract

We consider an opinion dynamics model coupled with an environmental dynamics. Based on a forward invariance argument, we can simplify the analysis of the asymptotic behavior to the case when all the opinions in the social network are synchronized. Our goal is to emphasize the role of the trust given to the environmental signal in the asymptotic behavior of the opinion dynamics and implicitly of the coupled system. To do that, we conduct a bifurcation analysis of the system around the origin when the trust parameter is varying. Specific conditions are presented for both pitchfork and Hopf bifurcation. Numerical illustration completes the theoretical findings.

Paper Structure

This paper contains 11 sections, 6 theorems, 36 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem formulation
  3. Preliminary results
  4. Forward Invariance of Synchronization manifold
  5. Oddness of the dynamics
  6. Analysis of synchronized agents dynamics
  7. Equilibria
  8. Singular points
  9. Bifurcation analysis
  10. NUMERICAL SIMULATIONS
  11. CONCLUSIONS

Key Result

Lemma 1

Let $\mathcal{G}$ be a connected graph. Then, the normalized adjacency matrix $\boldsymbol{D}^{-1} \boldsymbol{A}$ has a simple eigenvalue $1$, and all other eigenvalues have modulus strictly less than $1$. Moreover, the vector $\mathbf{1}$ is the right eigenvector associated with the eigenvalue $1$

Figures (3)

  • Figure 1: Bifurcation diagram for the FSO dynamics \ref{['eq:FSO_dynamics']} with $\gamma = 0.2$. Solid lines denote stable equilibria, dashed lines denote unstable equilibria, dots mark bifurcation points, and the green region shows the amplitude of the stable limit cycle emerging from the Hopf bifurcation.
  • Figure 2: Phase portraits for two value of $\beta$ and $\gamma = 0.2$. The system passes from 5 equilibria for $\beta = 0.24$ to 1 equilibrium with a stable limit cycle for $\beta =0.25$.
  • Figure 3: Phase portraits for two value of $\beta$ and $\gamma = 0.2$. The stable limit cycle for $\beta=0.59$ collapses into an equilibrium at $\beta = 0.61$ illustrating the Hopf bifurcation of Figure \ref{['fig:bifurcation_diagram']}.

Theorems & Definitions (14)

  • Lemma 1: Perron-Frobenius
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • ...and 4 more