From Dissensus to Consensus: Bias-Controlled Transition in Nonlinear Opinion Dynamics

We propose a novel bias-based consensus framework for nonlinear opinion dynamics. Due to the observable and malleable nature of bias in human-robot interactions, we utilize it as a control parameter to achieve consensus. First, we analyze the Lyapunov–Schmidt reduced system near equilibrium under small bias assumptions. Through constrained cusp bifurcation, we show that increasing individual biases beyond identified thresholds—and relative biases beyond saddle-node limit points ensures consensus with a unique stable equilibrium. For large biases, we conduct a global phase-plane analysis. By establishing strong monotonicity and applying the Poincaré–Bendixson theorem, we eliminate the possibility of limit cycles and guarantee consensus with a unique stable attractor as equilibrium. Finally, along with numerical simulations for the two-agent, two-option case, we show that the proposed bias control approach extends seamlessly to decentralized multi-agent opinion consensus.