TITLE:
A Continuous Dynamical Model of Collaborative Learning
AUTHORS:
George Dassios, Aggeliki Efstathiou
KEYWORDS:
Dynamical Systems, Coupled Systems, Stability Analysis, Consensus Dynamics, Nonlinear Coupling, Learning Dynamics
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.7,
July
17,
2026
ABSTRACT: We present a continuous-time model describing how two interacting individuals acquire, forget, and exchange knowledge during collaborative processes. The formulation extends the classical learning with decaying memory framework by introducing interaction terms that depend on the state difference between the two individuals. In the linear case, we derive the equilibrium and fully characterize stability, showing that interaction reduces state differences without affecting the mean level. A coupling threshold is identified, marking the transition from individual to collective dynamics. To reflect that interaction is effective only within a suitable range, we introduce a nonlinear, state-dependent coupling function. This leads to a convergence zone, where equalization is rapid, and a decoupling regime when differences are large. The model provides a tractable dynamical systems framework with potential applications to learning processes