Julia Dragons

The following provide Julia sets given by the dynamical system

zn+1= zn3 + zn + p

with various values of p plotted on the rectangular region with xmin = -1.1, xmax = 1.1, ymin = -0.7, and ymax = 0.7. Note that as the value of p decreases, the dragons get more obese.

Point of Mathematical Interest: If my computer can show any of the dragons by iterating the dynamical system indefinitely without using a maximum number of iterations M, then the picture will contain infinitely many black-to-red bands in its background. Topologically speaking, their union is an "open" region in the complex plane, and by removing it from the picture, we will be left with a so-called "compact" set of points whose boundary forms a Julia set. Strictly speaking, each dragon above is given by iterating the dynamical system at most M = 5,000 times for each point in the rectangle, so it only provides an “approximation” to a true Julia set. Nobody can distinguish such approximation and the true figure on a computer screen, however.

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