Seminar Details
2026-01-20 (14:00) : Opinion Dynamics with Nonlinear Interaction: From Robust Clustering to Environmental Coupling
At Euler building (room A.002)
Organized by Mathematical Engineering
Speaker :
Anthony Couthures (Université de Lorraine)
Abstract :
Social opinion and environmental states are deeply coupled: collective human behavior impacts the environment, while the state of the environment feeds back into public opinion. In this talk, I will present a mathematical framework to analyze these interactions, moving from standard consensus models to coupled socio-environmental dynamics.
First, I will introduce a generalized framework for multi-agent opinion dynamics with nonlinear interactions. Unlike classical linear consensus models, nonlinear communication allows for the emergence of rich behaviors beyond simple agreement. We will establish a sharp threshold linking network connectivity (algebraic connectivity) and interaction nonlinearity (Lipschitz constant) that dictates the transition from global synchronization to persistent polarization. Using Input-to-State Stability (ISS) theory, I will also discuss the robustness of these polarized clusters against external influence.
In the second part, I will couple this opinion model with an environmental resource variable. By analyzing the system on the synchronization manifold, we identify the role of the "attention parameter" β: representing the weight agents place on environmental feedback versus social influence. Through bifurcation analysis, I will demonstrate how varying this parameter triggers fundamental qualitative changes, specifically Pitchfork bifurcations (leading to bistability and polarization) and Hopf bifurcations (leading to recurrent cycles of environmental collapse and recovery).