For K-12 Students · Bifurcation Diagram
Bifurcation Diagram for a Markov chain
Built for k-12 students learning it in middle or high school. Watch the idea come alive with plain-language steps and everyday examples — perfect for projects and homework. Simulate a Markov chain live below — adjust the inputs and watch it respond, right in your browser.
Bifurcation DiagramLive
logistic map · route to chaos
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The logistic map xₙ₊₁ = r·xₙ(1−xₙ) doubles its period again and again as r grows, then dissolves into chaos near r ≈ 3.57 — with windows of order inside.
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Mapr·x(1−x)
Chaos onsetr ≈ 3.57
Feigenbaum δ4.669
Governing equation
Reading this result: Past r ≈ 3.57 the branches blur into chaos, yet pale vertical windows remain where periodic order suddenly returns.
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Frequently asked questions
- Is this good for k-12 students?
- Yes — this version of "Bifurcation Diagram for a Markov chain" is framed for k-12 students learning it in middle or high school. Watch the idea come alive with plain-language steps and everyday examples — perfect for projects and homework.
- Do I need to install anything?
- No. It runs in any modern browser, free, with no account required.