Periodic outbursts of social activity
In this talk I will discuss the existence and convergence properties of a continuous 2x2 dynamical system with time-periodic source term which is either of cooperative-type or activator-inhibitor type. This system was recently introduced in the literature to model the dynamics of social outbursts and consists of an explicit field measuring the level of rioting activity and an implicit field measuring the effective social tension. However, the system can represent any phenomena in which one variable exhibits self-excitement once the other variable has reached a critical value. The time-periodic source term allows one to analyze the effect which periodic external shocks to the system play in the dynamics of the outburst of activity. For cooperative systems we prove that for small shocks the level of activity dies down (approaching a quiet cycle) and as the intensity of the shocks increases the level of activity converges to a positive periodic solution (excited cycle). We further prove that in some cases there exist at least two positive periodic solutions. We provide a subset of these results obtained in the cooperative system for the activator-inhibitor system.