Logistic Map Calculator
Explore the logistic map x_{n+1} = r x_n (1 − x_n). This interactive tool computes orbits, displays cobweb plots, estimates Lyapunov exponents, and offers a bifurcation-preview feature. Use it for teaching, exploration, and quick experiments on chaotic dynamics.
Logistic map — a gateway to chaos
The logistic map, perhaps the most famous one-dimensional nonlinear map, offers an accessible route into modern nonlinear dynamics. Defined by the simple quadratic recurrence x_{n+1} = r x_n (1 − x_n), it displays a startling variety of behavior depending on the parameter r: stable fixed points, periodic cycles, period-doubling cascades, and chaotic regimes where long-term behavior becomes highly sensitive to initial conditions. This article explains the mathematical underpinnings, numerical exploration tips, and uses of the map in modeling and pedagogy.
Qualitative behavior as r varies
For small r (0 < r < 1), the population decays to zero. As r grows past 1, a non-zero fixed point appears and becomes stable for a range of r. At r ≈ 3 the fixed point loses stability and a period-2 orbit emerges — the start of the famous period-doubling route to chaos. Further increases in r cause successive period doublings (period-4, period-8, ...), accumulating at the Feigenbaum constant, beyond which chaotic behavior appears. Even within chaos there exist windows of periodicity (e.g., the period-3 window), making the map intricately structured.
Cobweb plots and visualization
Cobweb plots are invaluable for visual intuition. Plotting y = f(x) and y = x together, then drawing vertical and horizontal lines between them shows how an initial x0 moves under iteration. Convergence to a fixed point appears as a spiral into the intersection; cycles as closed polygonal loops; chaos as an erratic wandering pattern.
Lyapunov exponent and sensitive dependence
The Lyapunov exponent quantifies the average exponential rate of divergence of nearby initial conditions. Numerically compute λ by summing ln |f'(x_n)| along an orbit (after discarding an initial transient). A positive λ indicates chaos; λ ≈ 0 marks bifurcation thresholds; a negative λ indicates attraction.
Numerical best practices
When experimenting numerically, remove transients before sampling long-term behavior (discard first few hundred iterates for r in chaotic regimes). Use double-precision (browser Number) and be aware of rounding errors—chaotic orbits amplify tiny differences. For bifurcation diagrams and Lyapunov plots, sample r finely and collect post-transient iterates to visualize structure.
Applications and history
Originally introduced in population biology as a simple model with density-dependent growth, the logistic map gained fame in the 1970s as the study of deterministic chaos advanced. It serves as a teaching example in physics, mathematics, and engineering courses and a testbed for numerical methods in nonlinear dynamics.
Use this interactive page to explore and learn: vary r, inspect cobwebs, compute Lyapunov exponents, and experiment with bifurcation previews. For publication-quality figures and extensive studies consider specialized tools, but this calculator is ideal for intuition and classroom demonstrations.
Frequently Asked Questions
Typically 0 ≤ r ≤ 4. Values above 4 lead iterates outside [0,1] and eventual divergence to negative infinity for some x0.
Positive Lyapunov exponent means nearby trajectories diverge exponentially—typical of chaos and sensitive dependence.
Transient iterates reflect initial-condition effects. For long-term behavior and statistics, discard an initial transient to sample the attractor.
This tool provides a coarse preview. High-resolution bifurcation diagrams require many r samples and are computationally intensive.
Yes—slightly different x0 produce diverging futures. However, many x0 share the same invariant measure on the attractor in chaotic regimes.
Hundreds to thousands are fine for exploration. Avoid storing millions of iterates in the browser unless you stream or summarize them.
Yes—chaos is deterministic yet unpredictable due to sensitivity to initial conditions, often mistaken for randomness.
The code is focused on the logistic map but can be adapted to other one-dimensional maps with small edits.
Yes—use Download CSV to save iterates, Lyapunov sums and diagnostics for offline analysis.
Yes—AkCalculators provides this tool freely for education and exploration.