Not randomness
The same starting state and the same deterministic rule produce the same result.
INTERACTIVE SYSTEMS LABORATORY / EST. 2026
CHAOS
IS A VERB.
A butterfly is not a prophecy. A tiny change is not always a revolution. But when a system feeds back into itself, the future can become surprisingly hard to predict.
03 PLAYABLE EXPERIMENTS00 TRACKERS / ACCOUNTS∞ BETTER QUESTIONS
NOT A PERSONALITY TEST. NOT A FORECAST. JUST A MODEL YOU CAN OPEN.
01 / FIRST PRINCIPLES
Order can create surprise.
In everyday speech, chaos means disorder. In mathematics, deterministic chaos describes systems that follow rules yet become highly sensitive to starting conditions. Two almost identical beginnings can eventually produce very different paths.
Yesterday shapes today
Outputs become inputs. Small effects can accumulate, cancel, or oscillate.
Models aren't reality
A beautiful simulation can reveal a mechanism without predicting actual lives, markets, or ecosystems.
02 / THE LABORATORY
Touch one variable.
Watch the future fork.
Choose an experiment. Each runs entirely in your browser. No account, server computation or analytics.
THE BUTTERFLY EFFECT
Two populations follow the same logistic rule, with slightly different starting values. Increase the sensitivity parameter and watch their paths diverge.
COMPARISON DESK
Save a run, change a parameter, then compare outcomes. Nothing leaves your device.
WHAT YOU'RE SEEING
Run the experiment to compare two trajectories.
TWO FUTURES / SAME RULEMODEL / NOT FORECAST
PATH A PATH B
60 ITERATIONS · DETERMINISTIC
⌘ OPEN THE MODEL / ASSUMPTIONS
Logistic map: x(n+1) = r × x(n) × (1 − x(n)). Initial values: x₀ = 0.21 and x₀ + δ. Values are normalized, not measurements of real populations. Finite numerical precision matters.
03 / INTERPRET THE EVIDENCE
Chaos is not everything complicated.
The three machines above teach related but distinct mechanisms. The logistic map can display deterministic chaos; the threshold network models diffusion; the bounded feedback model illustrates multistability. Neither cascade nor tipping behaviour automatically demonstrates mathematical chaos.
Same rule. Different forecast horizon.
When the logistic map is in a sensitive regime, tiny errors in the starting state can become large differences later. Exact mathematical determinism does not grant unlimited practical prediction.
Try: Set growth to 3.20 and then 3.90 while keeping the starting difference unchanged. Compare final separation and the full curves—not only the last point.
Structure changes propagation.
Our 60-node ring is deliberately uniform: each node has four neighbours, adopts irreversibly, and follows one threshold. Real social networks are heterogeneous, and people can change their minds.
Try: Hold the seed at 15% and sweep the adoption threshold. Which transitions are abrupt? Would a different network topology behave differently?
History can matter.
The bounded feedback model can have two persistent states under identical external forcing. That is a model of multiple stable outcomes, not evidence that a real ecosystem or relationship is about to collapse.
Try: Increase feedback from 0.5 to 2.5 and compare both starting states. Which outcomes converge?
Four traps worth avoiding
Butterfly cliché: Not every small cause becomes a giant effect. Single-run certainty: A final point can hide earlier divergence and reconvergence. Analogy as proof: A mathematical example is not evidence about an actual person or society. Optimisation fantasy: A model can help ask better questions without prescribing one best intervention.
04 / THINKING CHALLENGES
Can you read a model without worshipping it?
01 · The weather forecast
Two numerical forecasts begin from slightly different estimates of the atmosphere and later disagree. Which conclusion is justified?
02 · A viral idea
A network simulation shows 90% adoption. What can you conclude about a real campaign?
03 · Two possible equilibria
One forcing input produces two long-run outcomes from different starts. What is the most careful interpretation?
05 / METHODS & LIMITATIONS
Open the black box.
Why these three models?
The logistic map isolates sensitivity to initial conditions in a one-dimensional recurrence. The ring network illustrates deterministic threshold diffusion. The hyperbolic tangent feedback rule illustrates how nonlinear reinforcement can support distinct persistent states. These are teaching models, not calibrated forecasts.
What would make the models more realistic?
Measured data, uncertainty intervals, alternative network structures, stochastic noise, varying thresholds, model validation and sensitivity analysis across many parameter combinations. More realistic is not always more understandable; assumptions should remain inspectable.
What is missing from our experiment?
We do not estimate Lyapunov exponents, test empirical systems, compute confidence intervals, simulate heterogeneous networks or establish bifurcation diagrams. The browser's finite-precision arithmetic also limits long-run interpretation.
How to test a claim rigorously?
Specify a falsifiable question, select measurable outcomes, vary one parameter while holding others constant, repeat over multiple initial conditions, compare rival explanations, and document when the analogy breaks.
06 / WHAT CHANGES?
Three ways to think more carefully.
Sensitivity is not destiny.
Some systems amplify differences. Others dampen them. The conditions matter more than the slogan.
A tipping point is not always obvious.
Gradual changes in a parameter can produce abrupt changes in behavior. But a model threshold does not establish a real-world threshold.
Intervention has trade-offs.
Changing one part of a network can propagate benefits, harms, and unintended effects. Ask whose outcomes count and who bears the uncertainty.
07 / YOUR FIELD NOTES
Keep the question.
Write one observation, one assumption you would test, and one place the model might fail. Your notes remain on this device unless you export them.
Nothing is uploaded. Notes are saved locally in this browser only.
09 / GO DEEPER
Read the foundations.
Four starting points, with different emphases. They are resources for learning, not evidence that every analogy on this page is scientifically established.