Journal of Psychiatry and Psychological Sciences

ABSTRACT

Dynamical systems approaches provide a useful framework for modelling interaction processes in multi-agent and relational settings. This paper introduces a minimal three-term force model combining (i) an oscillatory coupling term, (ii) a linear restoring component, and (iii) an inverse-square repulsive interaction from a background field of agents. The model is analysed both analytically and through numerical simulation.

The deterministic two-term field (oscillatory plus linear) is shown to possess a single stable equilibrium under the parameterisation studied (kem = 0.05, ω = 0.2, ρ = kg/kem = φ−1 ≈ 0.618), indicating that repulsive interactions are required to sustain non-trivial spatial structure. Numerical simulations of the full three-term system demonstrate that increasing repulsion strength α produces a continuous expansion in mean inter-agent separation, with no sharp phase transition or discrete stability window observed over the explored parameter range (0.002 ≤ α ≤ 0.025).

These results indicate that the system’s behaviour is governed by a smooth balance between attractive and repulsive influences rather than by a critical threshold. The model provides a compact and interpretable framework for studying how competing interaction terms shape collective organisation in simplified multi-agent systems. The golden ratio φ−1 appears in the model as a parameterised coupling ratio, not as an emergent attractor. Interpretations in relational or psychological domains are treated as analogical and remain outside the scope of empirical validation in the present work.

Keywords: Relational Dynamics; Dynamical Systems; Three-Term Force Model; Golden Ratio; Multi-Agent Coordination; Continuous Balance; Mathematical Modelling; Relational Psychopathology