The following provide Julia sets given by the dynamical system z_{n+1}= z_{n}^{3} + z_{n} + p with various values of p plotted on the rectangular region with |
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. |