Classification of dynamics for a two person model of planned behavior

Abstract

We study a dynamical system modeling the Theory of Planned Behavior (TPB) in which each individual's behavioral intention evolves continuously under an ODE driven by internal attitudes, perceived social norms, and perceived behavioral control. Actions occur as discrete threshold events: when intention reaches a fixed threshold it is reset to 0 and produces a transient "nudge" that jumps to 1 and then decays exponentially. This yields a hybrid ODE-threshold system with psychologically interpretable parameters. We derive a partial classification in the general case of n individuals. Focusing on the two-individual case (n=2), we obtain explicit formulas for trajectories between action events and derive bounds for first-action times. In the mixed setting where one individual is intrinsically increasing and the other is not, we identify a scalar invariant, M, measuring the net effect of one period of excitation. We prove that non-positive M is equivalent to a partial-action state (only the intrinsically active individual acts countable infinitely often), while positive M is equivalent to full action (both individuals act countably infinitely often). Finally, we demonstrate numerically that these analytic boundaries partition the parameter space with near-perfect agreement, and we provide exploratory simulations suggesting analogous structures for three individuals.