Computer Science Assignment | Mandelbrot Set Assignment

Added on - 28 Jan 2020

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(a)The Mandelbrot set is generated by what is called iteration, which means to repeat a processover and over again. In mathematics this process is most often the application of a mathematicalfunction. For the Mandelbrot set, the functions involved are some of the simplest imaginable:they all are what is calledquadratic polynomialsand have the formf(x) = x2+ c, wherecis aconstant number. As we go along, we will specify exactly what valuectakes.To iteratex2+ c, we begin with aseedfor the iteration. This is a number which we write asx0.Applying the functionx2+ ctox0yields the new numberx1=x02+cNow, we iterate using the result of the previous computation as the input for the next. That isx2=x12+cx3=x22+cx4=x32+cx5=x42+cand so forth. The list of numbersx0, x1, x2generated by this iteration has a name: it is calledtheorbitofx0under iteration ofx2+ cfor exampleSuppose we start with the constantc = 1.Then, if we choose the seed 0, the orbit isx0=0;x1=02+1=1;x2=12+1=2;x3=22+1=5;x4=52+1=26x5=262+1=677.....and we see that points in this orbit get bigger and bigger — the orbittends to infinity.As another example, choosec = 0. Now the orbit of the seed 0 is quite different: it remains fixedfor all iterations.x0=0;x1=0x2=0For c = -1, and seed 0, the orbit bounces back and forth between 0 and -1. This is acycle ofperiodx0=0;x1=02+i=i;x2=i2+i=1+i;
x3=(1+i)2+i=ix4=(i)2+i=1+ix5=(1+i)2+i=iand we see that this orbit eventually cycles with period 2. If we changecto2i, then the orbitbehaves very differently iex0=0;x1=02+2i=2i;x2=(2i)2+2i=4+2i;x3=(4+2i)2+2i=1214ix4=(1214i)2+2i=52334iFig Bitmap image from winfeedsoftwareThe Mandelbrot set therefore consistsof all of those (complex) c-values forwhich the corresponding orbit of 0under x2+ c does not escape to infinity.From our previous calculations, we seethatc = 0, -1, -1.1, -1.3, -1.38,andialllie in the Mandelbrot set, whereasc =1andc = 2ido notThe definition of the Mandelbrot setin theparameter plane.The behavior of the iteration sequenceZn+1:=fc¿¿) in thez-plane depends strongly on the valueof the parameterc.It turns out that for thosecsatisfying|c| > Rc, the set of pointszwhose iteration sequences donotconverge to infinity has area = 0
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