What was the aim of my study ?

The study of the fractals is very interesting
because a small modification of one parameter can lead to totally different
results.

Constants and data were consequently changed in order
to see this property.

First, I modified **X****0**and
**Y****0**

Then, the last of the simulation was
studied with the iteration number **n**.

Finally, I changed the constant **A,
B, C and D **untill I see a deep modification of the fractal image.

The King's Dream equation is the following one :

Xn = sin (Yn-1 * B) + C * sin (Xn-1 * B)

Xn = sin (Yn-1 * A) + D * sin (Yn-1 * A)

I took 3 different cases and I kept
the values of the main image.

Case 1 | Case 2 | Case 3 | |

X0 | 1 | 100 | 100 |

Y0 | 1 | 100 | 1 |

The following figure corresponds to case 2 because we have already the case 1 and the case 3 does not give more information.

We wonder that the figure is different but it is only a scale problem. Indeed, if I make a zoom, I have got the same graph as the case 1.

The initial values do not influence
the fractal because of the sinusoidal function like SIN.

The first plot is away from the final
image but the image is still the same.

We have the same behaviour with the
case 3.

The influence of the number of iterations n

I kept the same values as the main figure except n and I made 3 computations.

n=100 n=500 n=5 000

For n=100 there are some plots which
begin to make the fractal for n=500. If we have n=5000, it is a good picture
even if n=10 000 is more specific.

The right value is consequently equql
to 10 000.

The influence of A, B, C and D

Since the number of constants is such important, I decided to work with only one : A. The results are such identical for the B, C and D constants. I kept the same values as the main figure except A.

The main value of A is -0.966918 and I gave 5 values.

A=0 A=-0.5 A=-0.75 A=-0.85 A=0.90

If A is equal to zero, we can see like
a diffraction figure because Yn is equal to zero.

A=-0.5 gives us nothing which can
be considered interessant. We see only some plots.

When A is equal to -0.75, there is
a figure which is different from the main fractal. It is perhpas the beginning
of the main fractal.

A=-0.85 and A=-0.90 give the same
results than the main value of A. There is still the same figure
untill 10% of A.

The behaviour of King' Dream fractal
is consequently sensitive because after 10 % the fractal is different..