Classification of dynamics for a two person model of planned behavior

TL;DR

The paper analyzes a TPB-inspired hybrid system where behavioral intentions $x_i(t)$ evolve by ODEs and trigger discrete actions when crossing a threshold $\tau$, resetting to 0 and emitting a transient nudge $y_i$. Focusing on the two-agent case, it derives explicit inter-event trajectories, introduces a period-averaged invariant $M$ that separates partial from full action, and proves that $M\le0$ yields partial action while $M>0$ yields full action; a contraction mapping argument plus a special-case Lambert $W$ solution provide deeper insights. The results establish a complete analytical classification for the 2-person system and offer near-perfect agreement with numerical simulations, while suggesting analogous structures may extend to three agents. This work provides a rigorous mathematical bridge between threshold-driven action and TPB-like psychological dynamics, with potential applications to networked collective behavior and interventions in social systems.

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.

Figures (8)

• Figure 1: A simulated solution for a two individual case. Individual 1 (dark blue) has an inherently increasing behavioral intention and when this individual reaches the threshold, the behavioral intention is set back to zero, and a nudge (right panel) is created. This nudge increases individual 2's (light blue) behavioral intention momentarily, but not enough to cause individual 2 to act. After several actions by individual 1, individual 2 is finally incited to act, causing a nudge which increases individual 1's behavioral intention. This behavior will continue indefinitely
• Figure 2: The No action state in which both individuals have initially decreasing behavioral intentions
• Figure 3: The full action state in which both individuals have initially increasing behavioral intentions and so both will continue to act indefinitely
• Figure 4: Top an example of the initial data for individual 2's behavioral intention being very high and not being able to reach the behavior threshold. This behavior is qualitatively no different than the case on the Bottom where individual 2's initial behavioral intention is quite low.
• Figure 5: With each action of individual 1, individual 2's behavioral intention initially increases but decreases further than int increased by the next time individual 1 acts. This results in a net decrease for individual 2 and the solution is classified as partial action.

Theorems & Definitions (45)

• Proposition 1
• proof
• Proposition 2
• proof
• Corollary 1
• proof
• Lemma 1
• proof
• Corollary 2
• proof