Problem $2\mathrm{.}9$

BONUS. Benout B$.$ Mandelbrot, a famous mathematician is known as the inventor of fractals and this problem

is dedicated to him. In this problem you will generate the so$-$called quadratic Julia Sets, a well$-$known fractal

example.

Given two complex numbers, $c$ and $z_0$ the following recursion $($it is similar to fixed point iterations$)$ is defined

$z_n=z_{n-1}^2+c.$

For an arbitrary given choice of $c$ and $z_0,$ this recursion leads to a sequence of complex numbers $z_1,z_2,z_3,$dots

called the orbit of $z_0.$ Depending on the exact choice of $c$ and $z_0,$ a large range of orbit patterns are possible.

For a given fixed $c,$ most choices of $z_0$ yield orbits that tend towards infinity. $($That is$,|z_n|$ as $n).$

For some values of $c$ certain choices of $z_0$ yield orbits that eventually go into a periodic loop. Finally, some

starting values yield orbits that appear to dance around the complex plane, apparently at random $($an example

of chaos$).$ These initial values of $z_0$ make up the Julia set of this recursion, denoted by $J_c.$

Write a MATLAB$/$GNU Octave$/$Python script that visualizes a slightly different set, called the filled$-$in Julia

set denoted by $K_c,$ which is the set of all $z_0$ with orbits which do not tend towards infinity. The "normal" Julia

set $J_c$ is the edge of the filled$-$in Julia set. The figure below illustrates a filled$-$in Julia Set for one particular

value of $c.$

a$)$ It has been shown that if $|z_n|>2$ for some $n,$ then it is guaranteed that the orbit will tend to infinity. The

value of $n$ for which this becomes true is called the escape velocity of a particular $z_0.$ Write a function that

returns the escape velocity of a given $z_0$ and $c.$ The function declaration should be: $n=$EscVel$(z_0,c,N),$ where

$N$ is the maximum allowed escape velocity $($i$.$e$.$ if $|z_n|2$ for $NM=$JuliaSet$(z_{\text{M}\text{a}\text{x}},c,N),z_{\text{M}\text{a}\text{x}}{z}_{0}cNM{z}_{0}500500-z_{\text{M}\text{a}\text{x}}{z}_{\text{M}\text{a}\text{x}}-z_{\text{M}\text{a}\text{x}}{z}_{\text{M}\text{a}\text{x}}ZyZMMZN{z}_{\text{M}\text{a}\text{x}},$cNarctan$(0\mathrm{.}1**M)Myn,$ return $Nas$ the escape velocity, $so$ you

will prevent infinite loops