Global Hopf Bifurcation and Symmetric Periodic Solutions in Multi-Agent Systems with Neutral Distributed Delays

Abstract

We study the emergence of symmetric oscillatory behavior in multi-agent systems where each agent incorporates a continuous memory of its past states and past rates of change, modeled by distributed retarded and neutral delays. The closed-loop dynamics are described by a system of nonlinear neutral functional differential equations (NFDEs) with a high degree of symmetry, arising from a network of homogeneous agents. By reformulating the problem as a fixed point operator equation, we apply equivariant degree theory to establish rigorous conditions for unbounded global Hopf bifurcation from the consensus equilibrium. Our main results provide sufficient conditions for the local asymptotic stability of consensus and for the existence of unbounded global branches of non-constant periodic solutions with prescribed spatio-temporal symmetries. The question of whether such periodic solutions are stable (and therefore constitute periodic multiconsensus) is not resolved by the degree method; we address it in an illustrative example via numerical simulation. The example, which models eight coupled asset markets with momentum traders and fundamentalists, demonstrates how memory-driven instability can generate periodic boom-bust cycles across clusters of assets. The numerical experiments confirm the bifurcation predictions and reveal the stability of the resulting oscillations, illustrating the power of combining symmetric bifurcation theory with targeted numerical analysis.