Mandelbrot Hyperbolic Period Finder

The Mandelbrot Hyperbolic Period Finder is a lightweight analysis tool that takes a complex parameter $c$ as input (real and imaginary coordinates inside a minibrot) and then iterates $f(z)=z^2+c$ starting from the critical orbit, to look for an attracting cycle and estimate the corresponding hyperbolic component period, when one exists.

Overview

The Mandelbrot set is often explored visually, but hyperbolic components have a clean dynamical meaning: inside such a region, the map $f(z) = z^2 + c$ has an attracting periodic cycle. This tool looks for that cycle directly by iterating the dynamics of the quadratic map. If an attracting cycle is detected, the tool reports its period (the hyperbolic component period). If the point is near a boundary or associated with boundary features (e.g. Misiurewicz points or decorations), it may report uncertainty rather than a confident period.

Special note: The technical details that follow are based on information provided by the AI assistant used to develop this application.

What it does

Accepts a complex parameter $c = x + yi$ (manual input).
Iterates the quadratic map from the critical orbit ( $z_0 = 0$ ).
Searches for a repeating tail pattern consistent with a period $p$ .
Checks consistency via a p-step return test ( $f^p(z) \approx z$ ).
Estimates attraction strength using the cycle multiplier magnitude | $\lambda$ |.
Returns a period when the cycle is attracting (i.e., $|\lambda| < 1$ ).

What it is not

Not a renderer (no zooming, no coloring, no orbit plots).
Not a decoration classifier (visual “mini” look-alikes are out of scope).
Not a proof system: results are empirical and numerical.

Best used as a companion to visual tools like MandelScope, the Deep Zoomer, or other fractal generators. Explore visually there, then drop candidate points here to check the hyperbolic period.

What details it produces and what you can do with them

When a point lies inside a hyperbolic component, the app reports several dynamical invariants rather than visual features. The most important is the period of the attracting cycle, which identifies the type of hyperbolic component you’re in (period-2, period-3…, period-n). This is the defining property of a true mini Mandelbrot: it’s independent of scale, zoom level, or appearance.

Alongside the period, the app reports the cycle multiplier magnitude $|\lambda|$ , which measures how strongly attracting the cycle is. Smaller values generally indicate points deeper inside the hyperbolic component, while values closer to 1 suggest proximity to the boundary. This helps distinguish interior points, near-boundary points, and non-attracting boundary phenomena that can visually resemble minibrots.

The output may also include a confidence indicator and diagnostic statuses (such as boundary, inconclusive, or no cycle detected). These are especially useful around difficult points and help warn when a visually convincing structure is likely boundary-anchored rather than a genuine hyperbolic interior.

Practical workflow: explore visually, then copy a candidate coordinate, run it in the Hyperbolic Period Finder, and confirm whether it’s a true hyperbolic component. If it’s so, assign a consistent period label (e.g. “period-7 minibrot”) as the foundation for naming and cataloging.

Short taxonomy note: true minibrots vs. boundary look-alikes

In practice, several different kinds of “mini-looking” structures can appear in parameter space. This tool focuses on the dynamical distinction: does an attracting cycle exist?

True minibrot (hyperbolic component): Has an attracting periodic cycle for $f(c) = z^2 + c$ . The app returns “ok” and a well-defined period. Use this to label it structurally (e.g. “period-5 component/minibrot”).
Boundary / Misiurewicz-anchored decoration ( $\mu$ -island / decoration): Often looks “mini-like” visually, but does not have a stable attracting cycle at the anchor point. The tool may report “boundary” or “nonattracting”, “inconclusive”, or “no cycle detected”. Treat these as boundary phenomena rather than true hyperbolic interiors.
Near-boundary interior point: A point may be inside a hyperbolic component but very close to its boundary, making convergence slow. In that region, $|\lambda|$ tends toward 1 and the app may require more iterations or tighter tolerances to report “ok” with a stable period.

Naming tip: if you want reproducible names, start from the period (“period-n minibrot”), then refine with a location descriptor (e.g., “near seahorse valley”, “attached to period-3 bulb”, etc.) from your visual tool.

Interpreting results

The output is designed to be explicit about “what went right” and “what might be ambiguous.” Typical statuses include:

ok — Attracting cycle detected: Reports period, $|\lambda|$ , and a heuristic confidence score.
boundary_or_nonattracting — Cycle-like behavior found, but not attracting: Often near boundaries; candidate period exists, but $|\lambda|$ is $\approx 1$ or above.
inconclusive — Candidate period found, consistency check failed: The detected repeat pattern didn’t survive the stricter $f^p(z) \approx z$ test.
no_cycle_detected — No stable cycle detected within limits: Try increasing iterations or adjusting advanced parameters.
escaped — Orbit escaped: The point is outside the Mandelbrot set for these dynamics.

The “confidence” value is a heuristic stability indicator (not a probability). It blends attraction strength (|λ|), consistency error, and sensitivity to large periods.

Notes & limitations

Near the Mandelbrot boundary, convergence can be slow and numerically delicate.
Very large periods are more sensitive to floating-point noise.
Some boundary features may mimic periodicity without a truly attracting cycle.
If you get ambiguous output, try increasing maximum iteration and/or tightening “epsilon” carefully.