Sekino's Fractal Art Gallery 2D The gallery comprises fractal images that can be plotted on a plane by standard programming routines described in Stories about Fractal Plotting. We classify the images into four groups: The Mandelbrot Set Logistic Equation, Etc Julia Sets Newton Fractals § 2.  Logistic Equation and Other Dynamical Systems

Logistic Equation  zn+1 = p(1 - zn) zn The picture shown above is a "global" image generated by the logistic equation, which was created to model population dynamics. The equation was suddenly under the spotlight in the 1970s when its "chaotic" behavior was discovered and triggered the "Chaos Theory" to develop in mathematics. We can apply the logistic equation just like the Mandelbrot equation to find numerous fractal images. The first five images shown below are given by zooming in on the vicinity of the right-hand "antenna" of the global image, where the first sign of chaos was spotted in 1974.

Spring Reflection The image may be mapped onto:

Bowl and Apple Golden Autumn Early Summer Winter Night Circus Elephants Balancing Elephants Jumping Elephants If we zoom in on the midpoint between each pair of eyes in "Jumping Elephants," we find a "snowman" figure (a replica of the M set) and one of them is shown below. As we shall see, the "snowman" figure pops up frequently in many dynamical systems. And here is a "branch" from the M set. Continue or  Return to the Top:

Dynamical System  zn+1 = fp(zn) = zn3 + zn + p  In the global image of the third degree equation shown above at left, two "snowmen" are visible: One is shot by a Stone Age arrowhead and the other is the tiny figure on the right, whose magnified image is shown separately at right. Here are two scenes from the boundary of the tiny snowman:  We often use the above image on the left as a night sky in 3D landscapes such as the one shown below. We show more 3D images in Gallery II.

Mandelbrot Moon Other Dynamical Systems   There are infinitely many dynamical systems for us to play with and each one of them generates a unique "global" shape like the ones shown above. For example, the first image is given by  zn+1 = fp(zn) = p(1 + zn)(1 - zn) zn ,  and like the Mandelbrot set, it conceals numerous fractal images which we can find by zooming. A distorted or asymmetric global image like the one on the right is usually generated by using a noncritical point for the initial value.    Go to Top of the Page Fractal Home or
Gallery 2D: The Mandelbrot Set Logistic Equation, Etc Julia Sets Newton Fractals Gallery 3D: Fractal Mountains and Forests Fantasy Landscapes Fractals on Nonplanar Surfaces 