Beyond A Level
A simple recurrence can behave in surprisingly complicated ways.
The logistic map starts with a familiar idea: repeatedly applying a formula. As the growth parameter changes, the sequence can settle, cycle or become chaotic.
This makes it a powerful bridge from A level sequences and iteration into university-style dynamical systems.
Iteration lab
Iterate one simple rule and watch the behaviour change
The logistic map is x_next = r x (1 - x). It can settle down, oscillate, or become chaotic depending on the value of r.
r
3.720
Initial value
0.420
Long-run min
0.2436
Long-run max
0.9296
Bifurcation view
One parameter creates a whole landscape of behaviour
For each r-value, the diagram plots long-term values after the early iterations are ignored. Splitting branches suggest cycles; dense regions suggest chaotic behaviour.
Joy in the process
The point is not just to finish. It is to notice, test and return.
These tools are invitations to explore. A good mistake, a surprising pattern or a question you cannot yet answer is part of the work, not a failure of it.
The challenge is deliberate: the site should support thinking, not remove the need for it.
Future extensions