Cobweb Diagram Generator Calculator

Calculator Inputs

Iteration function f(x)

Use x and optional parameter r. Example: r*x*(1-x)

Parameter r

Initial value x₀

Iterations

x minimum

x maximum

Curve samples

Chart size in pixels

Example Data Table

Example settings: f(x) = 2.8x(1 − x), x₀ = 0.2, 6 iterations, viewing window from 0 to 1.

Iteration	xₙ	f(xₙ)	Interpretation
0	0.200000	0.448000	First jump rises toward the curve.
1	0.448000	0.692429	Second step moves closer to the stable region.
2	0.692429	0.596692	The path bends back toward the diagonal.
3	0.596692	0.673201	The sequence oscillates within a narrower band.
4	0.673201	0.616262	Alternating movement suggests attraction toward equilibrium.
5	0.616262	0.662784	Later steps continue settling near a fixed point.

Formula Used

A cobweb diagram studies the iterative process:

xₙ₊₁ = f(xₙ)

The generator draws the function curve y = f(x) and the identity line y = x. Starting from x₀, each iteration uses:

Vertical move from (xₙ, xₙ) to (xₙ, f(xₙ))
Horizontal move from (xₙ, f(xₙ)) to (f(xₙ), f(xₙ))

Fixed points satisfy f(x*) = x*. When the steps shrink toward a point, the sequence is converging. Repeating two or more values may indicate cycles. Expanding steps can indicate divergence or chaotic behavior.

How to Use This Calculator

Enter the iterative function in terms of x. You may also use parameter r.
Set the parameter value, initial value, number of iterations, and visible x-range.
Choose curve samples and chart size for smoother or faster rendering.
Click Generate Cobweb Diagram to place the result above the form.
Review the diagram, summary, and iteration table to inspect convergence or cycles.
Use the CSV button for data export or the PDF button for a printable report.

FAQs

1. What does a cobweb diagram show?

It shows how repeated application of a function transforms an initial value. The steps reveal convergence, divergence, oscillation, or periodic behavior by comparing the function curve with the line y = x.

2. Which functions can I enter?

You can enter algebraic and many standard functions using x, optional parameter r, and functions such as sin, cos, tan, sqrt, abs, log, exp, min, and max.

3. Why is the identity line important?

The identity line marks points where output equals input. Any intersection between y = f(x) and y = x is a fixed point candidate, which is central for studying long-term iteration behavior.

4. What does converging behavior mean?

Converging behavior means successive iterates move toward a stable value or repeating set. In the diagram, the step pattern contracts instead of expanding away from the diagonal or the curve.

5. Can this reveal periodic cycles?

Yes. If the iterates bounce between repeating values, the stepping pattern can outline a cycle. A common case is a period-2 orbit, where two values repeat alternately.

6. Why did my iteration stop early?

The calculator may stop early if the function becomes undefined, the values explode to very large magnitudes, or the difference between successive values becomes extremely small near a fixed point.

7. What range should I choose?

Choose a range that covers the important dynamics of your function and starting value. For logistic-map style examples, 0 to 1 is often the most informative window.

8. What is included in the exported files?

The CSV export contains the iteration table, including xₙ, f(xₙ), and change values. The PDF export captures the visible results section for printing or sharing.