Nonlinear Dynamics of Dyads with Self-Regulating Internal States and Nonreciprocal, Time-Decaying Interactions

We introduce a minimal model for emotional and relational interactions between two individuals. The framework consists of four coupled nonlinear differential equations: two describing the time evolution of the individuals’ emotional states and two describing the directed relational feelings between them. The model incorporates key mechanisms relevant to interpersonal dynamics, including internal emotional regulation, nonreciprocal influence, time-decaying interaction strength, coupling between (internal) emotional dynamics and (external) relational dynamics, and stochastic perturbations. To summarize the evolving dyadic state, we introduce a bounded scalar metric that quantifies overall emotional and relational alignment between the two individuals. We further analyze the qualitative behavior of the system using numerical simulations. The model is minimal and operates under explicit assumptions. Within these limits, however, it provides a compact mathematical framework for studying coupled emotional and relational dynamics at the level of a dyad, complementing mean-field approaches that focus on large populations.