Applied mathematics and modelling · Maths EE idea · Solid

From order to chaos in the logistic map

A research question to start from

How does the long-term behaviour of xₙ₊₁ = r xₙ(1 − xₙ) change with r, and where do the first period doublings occur?

A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.

Why it works as a maths EE

Exact stability calculations for fixed points and 2-cycles, then a controlled look at chaos.

Mathematics you would need

Iteration and fixed points
Stability using derivatives
Period-2 cycles
Bifurcation diagrams

Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.

One possible line of attack

Find fixed points and their stability.
Find the 2-cycle and where it becomes unstable.
Explore further doublings numerically and discuss.

Scope and difficulty

Solid. Solid.

Pitfalls

Chaos pictures with no calculations.
Undefined claims of 'chaos'.

Where to start reading

Search a library catalogue or a university's open lecture notes for: logistic map period doubling stability derivative; 2-cycle logistic map. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).

Make it your EE

Plan this EEOpens the free EE planner with this question as a starting point. Sharpen the questionSix quick checks for a focused research question. Self-check a draftCheck a section against each criterion, free.