The chapter is devoted to the analysis of possibilities in designing dynamic properties of avatars in multi-agent systems. It is shown that determination of “differential logic” of avatars, based on the use of differential equations, gives sufficient flexibility in describing their behavior over time. At the same time, the differential description usually involves a smooth response of the object to external or internal influences, which for avatars is usually not correct. To eliminate this weak point, it is proposed to use the technology of internal bifurcations, which allows to simulate discontinues effects in the avatar dynamics. It is shown that even when using relatively simple quadratic models of the Lotka-Volterra type, the technique allows to describe rather complex information interactions in the multi-agent systems.