Logistic Family

Explore the complex logistic map and the fate of its critical orbit. Read the guide.

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Fractal controls
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Desktop: right-click to switch between pan and point placement; scroll to zoom. Mobile: double-tap and hold the second tap to move a point; pinch to zoom. See the guide for details.

Dynamical plane

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Parameter plane

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Guide

Repeat the rule z → az(1 − z), allowing both z and a to be complex numbers. Moving a changes which starting points escape and which stay nearby.

Try this: right-click in the parameter plane and move the mouse around the gold dot, or double-tap and drag on mobile. Enable Show orbit to see repeated steps from the critical point z = ½.

Controls

  • Desktop: right-click a plot to switch between pan / zoom and point placement, independently for each plot. Drag to pan; in point placement mode the point follows the mouse. Scroll to zoom in either mode without moving the point.
  • Mobile: drag with one finger to pan. To move a point, double-tap, keep the second tap held, and drag; lift your finger to finish. Use two fingers to pan and pinch to zoom. Adding a second finger cancels point placement, so pinching cannot leave the root or parameter accidentally moved.
  • Orbit start: enable Show orbit, turn off its tracking/critical-point lock, and enable Move orbit start. Point placement in the dynamical plane then moves the orbit start instead of the parameter or root.
  • Fullscreen: use Fullscreen to enter; use the browser’s Back gesture on mobile or Esc on desktop to leave. There is no exit button covering the plot.
  • Keyboard: Tab to a plot, press M to switch modes, use the arrow keys to pan or move the point, + / − to zoom, and 0 to reset the view. Reset view also restores the initial pan and zoom.

Reading the picture

The dynamical plane fixes a and varies the starting point. The parameter plane varies a and follows the orbit starting at z = ½.

Colors encode escape time. Very dark pixels have not escaped within the iteration limit; increasing that limit can reveal more detail.

  • Gold dot: the selected parameter a, shown at the same coordinates on both panels.
  • Magenta dots: the fixed points 0 and 1 − 1/a (the second formula needs a ≠ 0).
  • Teal dot: the critical point z = ½ for a ≠ 0.
  • White path: an orbit when Show orbit is enabled. Turn off Lock to critical point and select Move orbit start to try another starting point.