The Three-Term Force Law

Let the state of agent i at time t be represented by a position vector xi(t) ∈ ℝ³. The total force

on agent i is:

Ftotal(xi) = kem sin(ω xi) − kg xi + α Σj rij / |rij|³

where rij = xi − xj and the sum runs over background repulsive agents (static field). The coupling ratio is defined as ρ = kg/kem. The gradient-flow structure of this system places it within a well- established class of dissipative dynamical systems studied extensively in nonlinear dynamics [8].

The φ⁻¹ Coupling Ratio as a Parameterised Operating Point

The coupling ratio ρ = kg/kem is set to φ−1 = (√5 − 1)/2 ≈ 0.618 throughout this study. This value is a chosen operating point, not a derived or emergent constant. The simulations do not establish φ−1 as a uniquely optimal value; a systematic sweep over ρ remains future work. No claim is made that the golden ratio has intrinsic significance for psychological or relational systems.

Dynamical Systems Context

Attractor states in dynamical systems models correspond to stable patterns toward which systems return after perturbation [1,8]. The force model proposed here does not exhibit sharp bifurcations over the explored parameter range; instead, it shows graded, continuous changes in mean inter-agent separation. This places it in a different dynamical class from catastrophe- theoretic models [2] that posit discontinuous transitions, while remaining formally related to the broader dynamical systems tradition in psychiatry [9]. The study of coordination dynamics has demonstrated how collective behaviour emerges from the interaction of simpler components; the present model extends this tradition by introducing a three-term force law with explicit parameterisation of the coupling ratio [10].

We begin by analysing the deterministic two-term component:

F(x) = kem sin(ωx) − kg x

Under the parameterisation kem = 0.05, ω = 0.2, kg = 0.618 kem, the linear restoring term dominates the oscillatory contribution over the domain examined (x ∈ [−2, 2]). The combined field exhibits a single stable equilibrium at x = 0. In the absence of repulsive interactions, the system collapses to a trivial configuration centred at the origin; the third term is therefore necessary to sustain spatial differentiation. Figure 1 illustrates the individual components and their superposition, confirming the single stable fixed point.

Continuous Balance as a Relational Analogy

The three-term model exhibits a continuous response to changes in α. Within the analogical mapping to relational dynamics, this suggests that relational configuration varies smoothly with ambient relational pressure. Low α (attraction-dominated) is interpretable as enmeshment or reduced relational differentiation 11. High α (repulsion-dominated) is interpretable as fragmentation or isolation [12]. Intermediate α corresponds to differentiated but coherent relational functioning.

The continuous nature of the response is the model’s central feature. Research on the transduction of social signals into physiological responses provides independent support for continuous rather than threshold models of relational stress [13]. The graded modulation observed in the present simulations is consistent with a view of relational dysregulation as a quantitative rather than categorical phenomenon. Dynamical systems approaches to social psychology have long emphasised continuous variation over discrete categories [14].

Implications for Multi-Agent Coordination

For distributed multi-agent systems, the model suggests a continuous coupling trade-off. At low α (tight coupling), agents cohere strongly at risk of redundancy. At high α (loose coupling), agents operate independently at risk of incoherence. At intermediate α, a balance between coordination and independence is achieved. The specific value α = 0.01 with ρ = φ ⁻¹ represents one point on this continuum not a uniquely optimal configuration. The study of self- organising pattern formation in behavioural systems has identified similar trade-offs between integration and segregation [15,16].

Parameterisation of ρ

ρ = kg/kem is fixed at φ−1. This is a chosen operating point, not an emergent constant. A systematic sweep over ρ remains future work.

Absence of sharp phase transitions

The mean pairwise distance metric varies monotonically with α. Regime labels are qualitative descriptors of gradual changes, not distinct phases.

Formal stability characterization

Lyapunov exponents and spatial variance measures are not computed.

Finite simulation scope

Simulations use finite particle counts, bounded domains, velocity damping, and soft reflecting boundaries. Sensitivity to these choices is not exhaustively characterised.

Static background field

The repulsive background agents are static rather than co-evolving.

Restricted parameter exploration

Only α is swept systematically

Illustrative trajectories

Figure 3 shows representative trajectories for a single random seed, not statistical summaries.

Interpretive scope

The mapping to psychological constructs is analogical, not empirical.

Absence of empirical validation

The study is simulation-based. No empirical data are incorporated.

If the continuous balance model bears analogical relation to psychological systems, several directions for empirical work follow:

These are directions for future research, not validated findings. Developmental dynamical systems research has similarly emphasised the importance of continuous measures in capturing behavioural change [18].

The present model joins a growing literature on mathematical approaches to psychiatric phenomena, including Hopfield network models, energy landscape models of depression, catastrophe theory models of therapeutic change, synergetic accounts of self-organising processes, and information-flow approaches to psychotherapy [1,2,13,14,16]. Unlike catastrophe- theoretic models, the present model exhibits graded, continuous changes. The two framings are complementary: continuous and discontinuous models may characterise different relational systems or different parameter regions. The model is closer in spirit to field-theoretic approaches to psychological modelling, with natural dialogue partners including phenomenological psychiatry’s accounts of intersubjective space and Stern’s present moment theory [12-14]. Research in nonlinear dynamical systems analysis for the behavioural sciences has similarly advocated for formal modelling approaches that capture continuous variation [19,20].