This is a non-linear dynamical systems model of water conflict in a fictional fragile context. It represents two forces — inter-community trust and conflict — that drive each other in cycles. It is built not to predict, but to demonstrate: complex social systems can produce persistent oscillation, path dependence, and sensitivity to timing in ways that linear planning tools cannot capture.
The model is a teaching tool, not a simulation of a real place. Every parameter and equation reflects a theoretical claim about how these forces interact, not an empirical measurement.
This is not "how much trust exists." It is the trust level at which conflict stops growing — the minimum trust needed for stability. When the orbit centre moves left on the horizontal axis, this threshold has dropped: the system is more resilient, needing less trust to keep conflict in check.
The frequency and intensity of active water disputes. This is the visible variable — the top of the iceberg. What evaluations measure. What log frames track. It oscillates continuously because the structural conditions generating it are never fully resolved — only cycled through.
The model is a Hopf bifurcation oscillator — a Van der Pol variant applied to a social system. Written in deviation coordinates u = T − Teq and v = C − Ceq:
The term μ·(R₀² − R²) is the limit cycle mechanism. When the trajectory is inside the target orbit (R < R₀), this is positive — it pushes the trajectory outward. When outside (R > R₀), it is negative — it pulls inward. Every trajectory, from any starting point, converges to the same closed orbit. This is what makes it a limit cycle rather than a centre.
ω is the oscillation frequency. μ is the damping strength — how quickly trajectories are attracted to the orbit. R₀ is the target orbit radius, determined by water stress and rights recognition.
In dynamical systems, an attractor is a set of states the system tends toward over time. This model has one attractor: the limit cycle — the closed orbit itself. Every trajectory spirals toward it and stays.
At the centre of the orbit sits an unstable equilibrium — technically a repeller. This is where dT/dt = 0 and dC/dt = 0 simultaneously. The system could theoretically stay here forever, but any perturbation sends it into orbit. It is marked as a purple dot on the phase portrait. It represents the theoretical "stable peace" that planning frameworks assume is achievable. The model says: you cannot reach that point. You can only change the size and centre of the orbit around it.
This contrasts with the previous two-attractor model, which had a stable peace fixed point and a conflict trap fixed point. That model asked: which basin does the system fall into? This model asks a different question: given that the system will always oscillate, how do we make the orbit smaller and the cycles less severe?
The orbit radius is determined by water stress buffered by rights:
The orbit centre (Teq, Ceq) is determined by the programme and rights parameters. Programme reduces conflict pressure, lowering Teq — the system needs less trust to stay stable. Rights reform strengthens how effectively trust suppresses conflict (raises d in Teq = c/d), also lowering Teq. Drought raises conflict pressure, raising Teq — more trust is required just to maintain the cycle.
Swings between very high conflict and very low trust. Communities experience severe recurring crises. Characteristic of: drought, absent rights, no coordination support. The orbit may push against the state-space boundaries, producing asymmetric cycles.
Oscillates within a narrow band. Conflict peaks are lower, trust troughs shallower. Communities can function even during conflict phases. Characteristic of: adequate water, recognised rights, active programme support working with the cycle rather than against it.
Because the system oscillates, when you intervene matters as much as what you do. The same intervention — activating the programme at 80% — produces different trajectories depending on where the system is in its cycle at the moment of activation.
The system is at the top of its orbit. Activating the programme shifts the orbit centre and shrinks R₀, but the system must first complete the remainder of the conflict arc before trust begins recovering. Things get worse before they get better. An evaluation at t+6 months may conclude the programme is failing — when it is simply poorly timed.
The system is at the bottom of its orbit. The same programme shift now works with the cycle's momentum. Trust continues rising, the orbit tightens, conflict peaks drop faster. An evaluation at t+6 months shows improvement. Same programme. Different timing. Completely different apparent outcome.
This is path dependence: the history of the system — specifically, where it is in its cycle at the moment of intervention — determines how the intervention propagates forward. A Theory of Change that does not account for cycle position will misattribute outcomes. It will conclude that a programme succeeded or failed based on timing, not quality. This is one of the most common sources of false learning in development evaluation.
This model is deliberately simple. It has two state variables and three parameters. It does not model individual agents, heterogeneous communities, spatial dynamics, political identity, institutional memory, or the specific history of Aravane. It produces one oscillating orbit per parameter set — not the complex multi-frequency dynamics of real conflict systems.
It is not empirically validated. The equations and parameters were calibrated to produce pedagogically useful behaviour: visible orbits, intuitive parameter sensitivity, meaningful scenario comparisons. The coefficients are theoretical claims, not measurements.
What it demonstrates — mathematically and rigorously — are three claims that are difficult to make with linear models: that some social systems oscillate rather than converge; that structural interventions change orbit geometry rather than just current state; and that timing of interventions matters because history shapes trajectory. These claims hold even when this specific model is a simplification.