Phase Portraits of a System of Two Interacting Actors

The article investigates the dynamics of various interactions between two partners described by a system of two nonlinear ordinary differential equations. The partners may be various social subjects, ranging from individuals and social groups to states and nations. The models are an extension of the Murray–Gottman model originally proposed to describe relationships between married people. New functions are introduced describing the own behavior of the actors in the absence of interactions as well as functions modeling the mutual influence of the partners. Phase portraits are constructed for systems with excitable, self-sufficient, and some other types of actors that do not strive to attain a neutral state, as in the Murray–Gottman model. New types of dynamic behavior are discovered. In particular, two nonlinear conservative models are proposed that demonstrate an oscillatory dynamic about the center. The models examined in this article demonstrate a rich set of phase portraits and may be applied to model various social interactions between two partners.