Lambda Fractal

Sur la droite réelle, l'application logistique est devenue un petit laboratoire du chaos. Laisse ses valeurs errer dans le plan complexe et elle s'ouvre sur la fractale Lambda, ou fractale λ. FractalPark utilise une forme centrée : c met à l'échelle deux facteurs complémentaires qui tirent dans des directions opposées.

Un changement de coordonnées affine rend cette formule conjuguée à l'application quadratique standard z² + c. Elle partage les traits de famille de l'ensemble de Mandelbrot, mais pas son contour : ici, un disque occupe la place du cardioïde, et les bulbes attachés s'organisent selon leurs propres périodes.

Ici, c tourne deux boutons à la fois — croissance et rotation. Fixe-le pour une vue de Julia, puis pousse-le doucement : un bassin presque rond peut devenir une frontière de branches et de spirales.

The map z → λz(1 − z) began as the complex extension of the real logistic map, studied extensively in the 1970s as a simple model of population dynamics and chaotic behavior. In 1980, Benoit B. Mandelbrot took the iteration to unrestricted complex λ and z. His paper, "Fractal Aspects of the Iteration of z → λz(1−z) for Complex λ and z," introduced the λ-plane parameter set—now the Lambda fractal—and placed it in the broader theory of iterated rational maps.

The groundwork was already there. Around 1917–1918, Pierre Fatou and Gaston Julia developed the general theory of iterated rational functions that later made families such as the logistic map available to explore. In 1986, Heinz-Otto Peitgen and Peter H. Richter brought the Lambda fractal to a wider audience in "The Beauty of Fractals," with some of the first widely seen computer-generated images of the set.

At full scale, the Lambda set begins with an almost circular body—the countershape to the Mandelbrot set’s heart-like cardioid. Smaller bulbs cling to the disk; each marks a stable periodic cycle, and the largest marks a period-2 cycle. Beyond them lie branching channels, tiny disk islands, and spirals.

The islands give away the difference: they are round here, not cardioid-shaped, because the algebra has changed. In Julia mode, c can select a filled disk or a many-armed vortex.