Teaching objectives
Can a fully deterministic system be unpredictable? In 1961, meteorologist Edward Lorenz discovered that the answer is yes: tiny differences in initial conditions produce radically different trajectories. This lab takes upper-secondary and university students to explore that phenomenon first-hand, moving from the Lorenz attractor to the double pendulum and population dynamics.
What you'll learn
Students practise quantitative observation, reasoning about models, and distinguishing periodic from chaotic behaviour using simulated and real data.
- Students observe the butterfly effect in the Lorenz attractor: three initially indistinguishable trajectories separate and diverge before their eyes.
- They manipulate the parameter σ to find the threshold at which the two wings of the attractor appear.
- With the double pendulum they experiment with mass and length ratios to see when chaos sets in earliest.
- They contrast the Malthus model (exponential) with the Verhulst model (logistic) using real world population data.
- They adjust the parameter k of the discrete logistic equation, distinguish periodic ranges from chaotic ones, and fit the curve to real caterpillar data.
- At the end they discover that chaos also generates structure: the bifurcation diagram of the logistic map is a fractal.
Key mathematical ideas
- Sensitivity to initial conditions: an arbitrarily small difference Δ grows exponentially over time.
- Lorenz attractor: the solution of a system of three nonlinear ODEs that traces an infinite orbit shaped like a butterfly.
- Discrete logistic equation: x_{n+1} = k · x_n · (1 − x_n); for k > 3.57 the behaviour is chaotic.
- Bifurcation diagram: shows how the period of the solution doubles successively until chaos is reached; its self-similarity makes it a fractal.
- Chaotic systems are deterministic but not predictable over the long term.
Room-by-room contents
Room 1 · Edward Lorenz and the butterfly effect
A narrative room that recounts how Lorenz discovered chaos in 1961 by rounding a number in his weather simulation. Introduces the idea of extreme sensitivity to initial conditions.
Student tasks
- Read the story of the discovery and move on.
Room 2 · Lorenz attractor — observe
Three trajectories start from nearly the same point on the Lorenz attractor and are animated in real time. At first they look like a single line; within seconds they diverge and trace different wings.
Student tasks
- Watch the animation and answer how many trajectories are visible at the start and how many after a few seconds.
Room 3 · Lorenz — separate after 15 s
The student controls the initial difference Δ between the trajectories. The challenge is to find a value of Δ such that, after exactly 15 seconds, the trajectories have clearly separated.
Student tasks
- Adjust the Δ slider and confirm the visible separation at 15 s.
Room 4 · Lorenz — the two wings
The parameter σ of the Lorenz system can be moved from 1. Below a certain threshold the trajectory collapses; above it the butterfly shape with two clearly distinct wings appears.
Student tasks
- Move σ upward from 1 and identify the value at which the attractor shows both wings.
Room 5 · Double pendulum — masses
A double pendulum simulation in which the student releases the system by dragging a weight. A button creates a copy displaced by a few degrees for trajectory comparison. The exploration variable is the mass ratio m₁/m₂.
Student tasks
- Release the pendulum and create the copy.
- Vary the mass ratio and observe which configuration makes the paths diverge fastest.
Room 6 · Double pendulum — lengths
Same mechanics as the previous room, but now the exploration variable is the length ratio L₁/L₂. The student looks for the configuration that maximises the speed of trajectory separation.
Student tasks
- Vary the length ratio and determine which configuration makes the paths diverge fastest.
Room 7 · Malthus, Verhulst and world population
A narrative-comparative room presenting real world population data alongside the exponential curve of Malthus and the logistic curve of Verhulst. The student decides visually which one it resembles more.
Student tasks
- Observe the real data and choose, with reasons, whether it looks more like the exponential or the logistic curve.
Room 8 · Logistic growth — explore
Discrete logistic map x_{n+1} = k·x_n·(1−x_n). The student varies k and observes how the curve transitions from converging to a fixed point, to periodic oscillation, and finally to chaotic behaviour.
Student tasks
- Move k and identify the range in which the curve is periodic.
- Locate the value of k from which the behaviour becomes chaotic.
Room 9 · Logistic map — real caterpillar data
Real data on annual caterpillar population oscillations. The student adjusts the logistic map sliders (k and initial population) to make the curve match the observations as closely as possible.
Student tasks
- Move the k and x₀ sliders to fit the curve to the caterpillar data.
Room 10 · A fractal born from chaos
Bifurcation diagram of the logistic map: for each value of k, the long-term stable values are plotted. The resulting image is a self-similar fractal. The student explores the structure by zooming in.
Student tasks
- Explore the bifurcation diagram.
- Identify the self-similarity by zooming into different regions.
Rooms to project
The most striking ones to show and discuss in class.