Calculator Inputs
Enter a recursive complex function of the form z(n+1) = z(n)p + c. The calculator studies orbit behavior, escape timing, and modulus growth.
Formula Used
Recursive generating function: zn+1 = znp + c
Complex constant: c = a + bi
Initial point: z0 = x + yi
Magnitude: |z| = √(real² + imag²)
Escape rule: orbit escapes when |zn| > R
Normalized escape estimate: ν = n + 1 − log(log|zn|) / log(p)
This model is common in polynomial fractal studies. Changing the power, starting point, and constant shifts orbit shape, stability, and escape speed.
How to Use This Calculator
- Enter the integer power used in the recursive function.
- Set the complex constant c using real and imaginary parts.
- Choose the starting point z₀ for the orbit.
- Define maximum iterations and an escape radius.
- Select the decimal precision for displayed results.
- Press Generate Fractal Orbit to calculate the sequence.
- Review the summary, orbit table, and graphs above the form.
- Use the export buttons to save the current orbit report.
Example Data Table
| Case | Power | c | z₀ | Escape Iteration | Final Modulus | Status |
|---|---|---|---|---|---|---|
| Classic quadratic Julia-style orbit | 2 | -0.7000 + 0.2702i | 0.0000 + 0.0000i | Not reached | 0.692797 | Bounded |
| Cubic orbit with gentle drift | 3 | -0.1200 + 0.7400i | 0.1000 + 0.0500i | Not reached | 0.593316 | Bounded |
| Quartic orbit with faster divergence | 4 | 0.2800 + 0.0100i | 0.2000 -0.1000i | Not reached | 0.286908 | Bounded |
Frequently Asked Questions
1) What does this calculator measure?
It tracks the orbit produced by a recursive complex function. The tool reports modulus growth, escape timing, final position, and a visual path through the complex plane.
2) What is the generating function here?
The page uses zn+1 = znp + c. This is a standard polynomial recurrence used to study many Julia-style and escape-time fractal behaviors.
3) Why do I enter real and imaginary parts?
Fractal orbits usually evolve in the complex plane. Splitting complex numbers into real and imaginary components makes the calculation easier to enter and interpret.
4) What does escape iteration mean?
It is the first iteration where the orbit magnitude becomes larger than the selected escape radius. Earlier escape often signals faster divergence from bounded behavior.
5) Why might an orbit stay bounded?
Some constants and starting points produce values that remain within the chosen radius for every tested step. That suggests local stability, though it is limited by your iteration count.
6) What does the normalized escape value show?
It refines the raw escape iteration using the orbit magnitude at escape. This helps compare nearby points with smoother grading than whole-number iteration counts alone.
7) What graph is shown after calculation?
You get a complex orbit plot and a modulus-by-iteration graph. Together they show directional movement, clustering, divergence, and overall growth of the sequence.
8) Can I export the results?
Yes. The CSV file stores the orbit table. The PDF file saves a compact report with summary metrics and orbit rows from the current calculation.