Maths IA · Modelling · topic 90 of 97
Chaos: does a butterfly change the weather? · IB Maths IA idea: guide, sample graph and step-by-step plan
no randomness, still unpredictableStart two simulated weather systems almost exactly the same. Wait. They stop looking alike. Can you measure how fast they drift apart, and how far ahead any forecast can be trusted?
- ✓ you like code and want to see a famous equation misbehave
- ✓ you enjoy asking how much a numerical answer can be trusted
- ✓ you want a topic where a straight line on a log graph tells you something deep
The big idea
Lorenz boiled a model of rising warm air down to three coupled differential equations. They have no closed-form solution, so you simulate them. The path never repeats and never settles, but it stays on a butterfly-shaped set. Two paths that start a tiny distance apart separate roughly exponentially at first, so the log of their distance against time is close to a straight line, and its slope measures how chaotic the system is. From that slope you can estimate how long a forecast stays useful. The simulation is the routine part. The marks come from separating what the equations do from what your numerical method does: compare Euler's method with a better one, change the step size, and show which differences are real.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Run Euler's method and a fourth-order Runge-Kutta method on the same start and compare the paths.
→ A numerical method becomes the thing you test. Find how small the Euler step must be to stay close for a given time.Start two paths a tiny distance apart and plot the distance between them against time.
→ A graph on a log scale. Explain the straight stretch, and why the distance stops growing.Fit a line to the log of the distance and use its slope as an estimate of the largest Lyapunov exponent.
→ A regression with a clear physical meaning. Repeat for many start points and give the spread.Use the slope to find how long it takes a tiny starting error to grow to a size you choose.
→ A logarithm solved for time. Show why a thousand times better measurements buy only a few more time units.Vary the middle parameter ρ and see when the motion settles to a fixed point, repeats or turns chaotic.
→ A family of behaviours. Plot a long-run quantity against ρ and describe where it changes.Repeat the idea with the logistic map x → rx(1 − x) and compare how the separation grows.
→ Chaos without differential equations. Compare the two growth rates and say what each tells you.Your Maths IA research question: run the check
which number will you measure?Write your own question first. Then make sure it ticks every box:
- Names the system and its parameters, and says the equations are used as a model, not as real weather.
- Names the quantity you will find: the growth rate of separation, the forecast horizon, or the step size needed.
- Says how the numerical method is checked and how the step size was chosen.
- Goes beyond running code: there is a log graph with a fitted line and a calculation from its slope.
- States what you leave out: the real atmosphere, measurement noise and the limits of the three-equation model.
The model
Three coupled equations and two paths that start almost together. Sketch the pair of paths for your own start point and mark where they part.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why this system, what you expect before simulating: do two close starts stay close?
Each equation and parameter defined, the origin of the model in a sentence, and the fixed points found by solving.
Euler's method and a better one written out for three equations, with a hand check of the first step.
The distance between two paths against time, on a log scale, with the straight stretch identified.
A fitted line, its slope, its spread over many start points, and what it means.
The time for a starting error to reach a chosen size, with a table for several error sizes.
Back to your aim: the growth rate, the horizon and how the numerical method changes both.
Step size, method, window for the fit and the limits of the model. Say which choice affects the answer most.
Where the equations, parameters and methods came from.
What your Maths IA graph could look like
where does the line stop being straight?Made-up readings to show the form of a strong graph, not the answer: the log of the separation of two paths against time, with a fitted straight line. Yours will look different. example only
- The log of the separation against time, so exponential growth shows as a straight line.
- Axis labels give units: model time units on the time axis, a log of a distance (no unit) on the other.
- Choose the fitting window before you look at the slope, and say how you chose it.
- Plot results from several start points: the scatter shows how much to trust one slope.
- Another graph to consider: the path in the x-z plane, to show both lobes of the attractor.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out what 'sensitive dependence' means. Before any code, guess how long two starts a billionth apart stay close.
- Set up the systemWrite the three equations and the standard parameters, and say what each term does. Define every symbol.
- Trial run (1 lesson)Code Euler's method for one start point and check the first three steps by hand. Fix any mismatch now: signs, step size, order of updating.
- Add a better methodCode a fourth-order Runge-Kutta step. Compare it with Euler at several step sizes and choose a step you can defend.
- Run two startsStart two paths a tiny distance apart, record the distance each step and plot its log against time.
- Fit the growth ratePick the straight stretch, fit a line and read the slope. Repeat for many start points and give a spread.
- Find the horizonUse the slope to find the time for a given starting error to reach a chosen size. Test it against your simulation.
- Go further, with a reasonVary a parameter, or repeat with a simpler chaotic map, and say what the comparison shows. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it says about real forecasting and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Coupled differential equations – Write , , and find the fixed points by solving them simultaneously.
- Euler's method – Step all three variables together and measure how the step size changes the path and the error.
- Exponential growth and logarithms – Model the separation as and take to get a straight line of slope .
- Linear regression – Fit a line to against over a chosen window and give the slope and its spread across start points.
- Solving for a time – Solve for the forecast horizon and tabulate it for several .
Key equations
Syllabus links
- 1.5Exponents and an introduction to logarithms
- 2.9Exponential and logarithmic functions
- 4.4Correlation and the regression line of y on x
- 5.18First-order differential equations; Euler's methodHL
- 2.10Log scales, linearising data, log-log and semi-log graphsHL
- 4.4Correlation and linear regression
- 5.1Limits and the derivative as a rate of change
- 5.16Euler's method, including coupled systemsHL
Read up first
- Coupled differential equations (AI HL) →
- Differential equations (AA HL) →
- Exponentials and logarithms (AA HL) →
Data and sources to start from
Simulated paths of the Lorenz system, computed by the student with Euler and Runge-Kutta methods; no external data is needed.
References to cite
- Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2), 130-141.
- Strogatz, S. H. (2015). Nonlinear dynamics and chaos (2nd ed.). Westview Press.
- Gleick, J. (1987). Chaos: Making a new science. Viking.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
test your step size- Calling chaos randomness. The equations have no randomness: the same start always gives the same path.
- Using Euler with a large step and treating the result as the system's behaviour. Test the step size.
- Fitting a line through the whole log graph. It flattens when the paths are as far apart as the attractor allows.
- Quoting one slope from one start point. The growth rate varies along the path, so give a spread.
- Taking a forecast horizon as exact. It depends on the starting error and on the size you call 'wrong'.
- Saying the model is the weather. It is a three-equation caricature of one effect.
More Maths IA ideas like this
Lynx and hares: predator-prey cycles●●●
Newton's method: how fast, and when does it fail?●●○
The Mandelbrot set: which numbers stay bounded?●●●
Epidemics: does a top-down model match a real outbreak?●●●
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.