  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 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 is
introduced as a time scaling, depending on the overall magnetization of the
system, 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, which is in accordance with the one
obtained by simulating the N-particle system.
