Bifurcation Diagram
Classroom worksheet · Interactive simulation: https://lkforge.com/tools/physics/bifurcation-diagram/
Name: Date:
Learning objectives
- Describe the logistic map and its long-term behavior.
- Identify period-doubling as a route to chaos.
- Locate where chaos begins.
Variables to change
- Growth parameter r
- Initial population
Procedure
- Set a low r and observe the population settle to a single steady value.
- Increase r and watch it split into 2, then 4, then more values.
- Increase r past about 3.57 and observe chaotic behavior.
Observations
Record the number of stable values as r increases and where the pattern turns chaotic.
Questions
- What does each branch split (bifurcation) represent?
- What is the logistic map equation?
- Roughly where does chaos begin?
- Is the chaotic region completely disordered?
- What does this simple equation demonstrate?
Answer key (instructors)
- 1. A period doubling — the population now cycles between twice as many values.
- 2. x_{n+1} = r·x_n·(1 − x_n).
- 3. Around r ≈ 3.57, after an infinite cascade of period doublings.
- 4. No — it contains narrow "windows" of periodic behavior (e.g. period-3) amid the chaos.
- 5. That very simple deterministic rules can produce complex, chaotic behavior.
The logistic map x_{n+1} = r·x_n·(1−x_n) settles to one value at low r, then period-doubles (2, 4, 8, …) as r rises, reaching chaos near r ≈ 3.57. This simple rule shows how deterministic systems can become chaotic through period doubling.