A story about conflict, polarisation, and what a simple equation reveals about why peace is so hard
In 2007, a team of psychologists at Columbia University made an uncomfortable claim. They argued that some conflicts — the ones we call intractable, the ones that resist every intervention — are not political failures, or failures of leadership, or failures of will. They are, in a precise mathematical sense, attractors.
Peter Coleman, Robin Vallacher, and Andrzej Nowak had been studying dynamical systems theory and applying it to protracted social conflict. What they found was that the same mathematics used to describe populations of fish, or the oscillations of a pendulum, described the behaviour of societies locked in violence.
This is an exploration of that idea — using the simplest nonlinear equation in existence.
Start with a community. Imagine measuring, each month, what fraction of people hold a hostile attitude toward an outgroup — a number between 0 (no hostility) and 1 (total hostility). Call it x.
Now ask: what determines how that number evolves over time? Two forces are in tension. Social reinforcement — the more people around you hold a view, the more likely you are to adopt it. And a natural ceiling — a society cannot sustain hostility indefinitely at full intensity; exhaustion, social cost, and dissenting voices push back.
This tension produces a structure that is formally identical to the logistic map:
This is not a metaphor. Nowak, Szamrej & Latané (1990) formalised exactly this structure in their dynamic theory of social impact — showing that public opinion evolves under reinforcement and resistance in ways that mirror population dynamics. The logistic map is not borrowed from biology. It is the natural shape of any process that grows under constraint.
What changes is what r means. In a fishery, r is reproduction rate. In a community, r is the intensity of social reinforcement — how strongly people's attitudes are shaped by those around them. The question is: what happens as r increases?
In a community with moderate social reinforcement — r below 3 — the system settles. Hostile attitudes may rise and fall in the short term, but the system converges to a stable level. Dissenting voices have room. Different social circles moderate each other. The conflict exists, but it is contained.
This is what Coleman calls a fixed-point attractor for conflict: a state that the system gravitates toward and, when disturbed, returns to. It is not peace — but it is manageable. Intervention is possible. Peacebuilders have traction.
Coleman, P.T., Vallacher, R., Nowak, A., & Bui-Wrzosinska, L. (2007). Intractable conflict as an attractor: Presenting a model of conflict, escalation, and intractability. American Behavioral Scientist, 50(11), 1454–1475.
Now imagine the social reinforcement pressure increases — through social media algorithms, charismatic leaders who sharpen us-versus-them narratives, or economic shocks that intensify group competition. r crosses 3.
Something structural happens. The fixed point becomes unstable. The system no longer settles — it begins to oscillate between two states. One month, hostility is high. The next, it retreats. Then surges again. Not randomly — in a structured cycle.
This is a bifurcation: a qualitative change in system behaviour triggered by a quantitative change in a parameter. The same community, the same people — but a different regime. The conflict is no longer a level to be managed. It is a cycle to be escaped.
As r increases further, the cycle doubles: two states become four, become eight. Each doubling happens faster than the last, compressing into an accumulation point — and then something more radical happens.
Beyond the accumulation point — at approximately r = 3.57 — the period-doubling cascade ends and something qualitatively different begins: deterministic chaos. The system is still governed by the same equation. There is no randomness. But the behaviour becomes sensitive to initial conditions in a way that makes long-range prediction impossible.
This is the formal structure of what Coleman and colleagues describe as intractable conflict. The system is not stuck in one bad state — it is locked in an attractor that constrains the space of possible states without settling into any of them. Conflict and apparent calm alternate unpredictably. Ceasefires fail not because of bad faith but because they are perturbations in a chaotic regime — they push the system away from one trajectory, but the attractor pulls it back.
Vallacher et al. (2010) note that in the chaotic regime, small interventions can have amplified effects — for good or ill. This is not a reason for pessimism. It is a map.
Vallacher, R.R., Coleman, P.T., Nowak, A., & Bui-Wrzosinska, L. (2010). Rethinking intractable conflict: The perspective of dynamical systems. American Psychologist, 65(4), 262–278.
The bifurcation diagram below shows the entire story in one image. For each value of r — each level of social reinforcement pressure — the diagram shows which states the system eventually occupies. One point means a fixed attractor. Two points means oscillation. A cloud means chaos.
Three features are worth noting. First, the transitions are sudden: the system doesn't gradually become more unstable — it shifts regimes at a bifurcation point. Second, the bifurcation points are coloured by regime: green for stable, gold for cycling, red for chaos. Third, there is a narrow band of gold inside the red — the period-3 window around r ≈ 3.83 — a small attractor of order that survives inside a chaotic landscape.
Coleman and colleagues call the equivalent of this window a latent attractor for peace: an alternative stable state that co-exists with the destructive attractor, waiting to be activated.
The vertical dashed lines mark the Feigenbaum bifurcation points — the precise values of r where each period-doubling occurs. Their spacing follows a universal law.
In 1975, physicist Mitchell Feigenbaum noticed something strange. The ratio between successive bifurcation intervals was converging to a constant — 4.6692… He then checked whether this constant was specific to the logistic map. It wasn't. It appeared in every smooth unimodal map — the sine function, the quadratic family, any system with this shape of nonlinearity.
The implication is significant: the route to chaos is universal. The specific mechanism — fish reproduction, opinion dynamics, market prices — doesn't matter. The structure of how order breaks down follows the same law.
What this means for peacebuilding: the dynamics of conflict escalation are not idiosyncratic to any particular conflict. There is a structural grammar to how systems move from stability to oscillation to chaos — and that grammar is knowable.
If conflict dynamics follow nonlinear attractors, then the standard toolkit of peacebuilding — rational dialogue, information provision, confidence-building measures — is optimised for a regime that may not be the one you're in.
Coleman, Vallacher and colleagues propose three strategies that follow from the dynamical perspective. They are worth reading against the bifurcation diagram above.
Intractable conflicts are locked in an attractor partly because complexity has collapsed into a single dimension — us vs them. Restoring complexity to the conflict landscape — surfacing cross-cutting issues, non-aligned actors, ambivalent identities — is the first step to destabilising the attractor. On the bifurcation diagram: reducing r.
The period-3 window in the chaotic regime — a small island of order — is the mathematical analogue of what Coleman calls a latent peace attractor: a stable alternative state that coexists with the destructive one. Peacebuilding is, in part, the work of building and strengthening these latent states before they are needed.
Coleman, Vallacher & Bartoli's analysis of Mozambique found that the Community of Sant'Egidio succeeded partly because it operated with low coercive pressure — weak power. Strong interventions in a high-r system often reinforce the attractor. Small, well-placed perturbations can shift the system to a different basin of attraction.
Morrison, Kutz & Gabbay (2022) showed that as a system approaches a bifurcation — a tipping point toward war or peace — it exhibits critical slowing down: perturbations take longer to recover from. This is a detectable early warning signal, visible in data before the transition occurs.
John Paul Lederach, whose moral imagination framework has shaped a generation of peacebuilders, argues that transformative moments in peacebuilding resemble aesthetic rather than technical acts — moments when something "intellectually and emotionally complex is captured in an 'ah-ha' moment." The dynamical systems perspective gives that intuition a formal structure: transformation happens at bifurcation points, and recognising when a system is near one is a skill that can be cultivated.