Oscillatory Behavior in a Model of Non-Markovian Mean Field Interacting Spins

We analyze a non-Markovian mean field interacting spin system, related to the Curie–Weiss model. We relax the Markovianity assumption by replacing the memoryless distribution of the waiting times of a classical spin-flip dynamics with a distribution with memory. The resulting stochastic evolution for a single particle is a spin-valued renewal process, an example of a two-state semi-Markov process. We associate to the individual dynamics an equivalent Markovian description, which is the subject of our analysis. We study a corresponding interacting particle system, where a mean field interaction-depending on the magnetization of the system-is introduced as a time scaling on the waiting times between two successive particle’s jumps. Via linearization arguments on the Fokker–Planck mean field limit equation, we give evidence of emerging periodic behavior. Specifically, numerical analysis on the discrete spectrum of the linearized operator, characterized by the zeros of an explicit holomorphic function, suggests the presence of a Hopf bifurcation for a critical value of the temperature. The presence of a Hopf bifurcation in the limit equation matches the emergence of a periodic behavior obtained by simulating the N-particle system.