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.

Logistic Map Bifurcation

x_{n+1}=rx_n(1−x_n): scan r, plot attractors — period doubling to chaos (Feigenbaum cascade).

About this model

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

xₙ₊₁ = r xₙ(1 − xₙ), 0 ≤ x ≤ 1

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.

Continue with similar topics in this category — or all 85 in Math Visualization.

View category →

NewUniversity / research

Rössler Attractor

Launch Simulator

RK4 integration of ẋ=−y−z, ẏ=x+ay, ż=b+z(x−c); period-doubling cascade as c grows.

NewUniversity / research

Lorenz Strange Attractor

Launch Simulator

σ, ρ, β ODEs; sensitive butterfly in (x,z) projection — RK4 trace.

NewUniversity / research

Three-Species Food Chain (Hastings–Powell)

Launch Simulator

Plants x → herbivores y → predators z; Holling II fᵢ(u)=aᵢu/(1+bᵢu); logistic x; chaotic attractors when b₁ is varied (1991).

NewKids

Chaos Game (Sierpiński)

Launch Simulator

Random vertex + midpoint walk; the attractor is the Sierpiński gasket — try RGB by vertex.

NewUniversity / research

Minkowski Diagram

Launch Simulator

Light cone and boosted axes in 1+1D; γ from v.

NewUniversity / research

Kalman Filter 2D Tracking

Launch Simulator

4-state constant-velocity tracker: noisy (x,y) measurements, process acceleration noise Q, measurement variance R, and a live ≈2σ covariance ellipse.

Logistic Map Bifurcation

About this model

How it works

Key equations

Frequently asked questions

Related simulators

Rössler Attractor

Lorenz Strange Attractor

Three-Species Food Chain (Hastings–Powell)

Chaos Game (Sierpiński)

Minkowski Diagram

Kalman Filter 2D Tracking

Logistic Map Bifurcation

About this model

How it works

Key equations

Frequently asked questions