Logistic Map Bifurcation
The logistic map x_{n+1} = r x_n (1 - x_n) is the simplest famous example of deterministic chaos in discrete time. For each parameter r between about 2 and 4, orbits stay in [0,1] (for typical initial conditions). When r is small, the map has a stable fixed point; as r increases, that fixed point loses stability in a period-doubling bifurcation, then the 2-cycle loses stability to a 4-cycle, and so on — the Feigenbaum cascade. Beyond an accumulation point r_∞ ≈ 3.569945…, chaotic bands appear interleaved with periodic windows. This simulator draws a bifurcation diagram: for many values of r along the horizontal axis, it discards transient iterations, then plots subsequent values of x vertically. Brighter pixels mean the orbit visits that (r, x) region more often. It complements the continuous logistic growth simulator (dN/dt = rN(1 - N/K)), which has no chaos in its scalar autonomous form.
Who it's for: Undergraduate students in dynamical systems, chaos, or numerical modeling; anyone comparing discrete vs continuous logistic models.
Key terms
- Logistic map
- Bifurcation diagram
- Period doubling
- Feigenbaum constant
- Deterministic chaos
- Attractor
- Orbit diagram
How it works
Discrete logistic map xₙ₊₁ = r xₙ(1 − xₙ) on [0, 1]: each column is a value of r; after discarding transients we plot many iterates of x. You see the period-doubling cascade into chaos; at fixed r a vertical slice is like an orbit diagram. Pairs with the continuous Logistic Growth lab (dN/dt).
Key equations
Frequently asked questions
- Why does the picture look fuzzy or thick in the chaotic region?
- Chaotic orbits are sensitive to initial conditions and explore a structured invariant set (often a Cantor-like attractor in slices). Plotting many iterates fills out the vertical extent of that attractor. Fine transients and finite iteration counts also add thickness.
- What is the dashed vertical line at r ≈ 3.57?
- It marks the approximate accumulation point r_∞ of the period-doubling cascade — beyond this, period-2^n cycles have all destabilized and more complex (often chaotic) dynamics dominate, though periodic windows still exist inside the chaos.
- Is this the same model as the Logistic Growth simulator?
- They are related but different: Logistic Growth integrates the continuous ODE dN/dt = rN(1 - N/K). The logistic map is a discrete-time recurrence with similar-looking nonlinearity; it can show bifurcations and chaos that the scalar ODE does not.
- Why change “Transient steps” and “Plotted iterates”?
- Transients are iterations discarded so you see long-term behavior (the attractor) rather than the approach from x0. More plotted iterates fill in the diagram more densely but cost more compute; if transients are too small, you may see ghost curves from not-yet-converged behavior.
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