2 Set of Julia
Definition

The full set of Julia Kc is defined as: The set of Julia is composed of all the initial values (z0) that entails the convergence of the chain previously defined. The border of this field is called set of Julia Jc.
With these definitions, the chaos can be geometrically observed. Indeed, if a point is near the border of the set of Julia but not in it, the chain beginning by this point will diverge. The extreme sensibility of the initial conditions can then be observed.

Plotting method

To plot the results, one method is to take each point of the calculation field as the initial value z0 for a given c and compute the chain. If there is convergence of the chain - which means that after a fixed number of iterations, |z| is inferior to an arbitrarely fixed number M - the initial value is plotted. The form we get depends on the value of c.

Matlab program

Program: julia.m

Results

• The rabbit of A. Douady
c is chosen for having 0 cyclic with an order of 3: 0 gives c, c gives c^2 + c and c^2 + c gives (c^2 + c)^2 + c. So c is solution of the equation:     (c^2 + c)^2 + c = 0. The value taken for c was an approximation: c = -0.1 + 0.7i.
The cycle {0, c, c^2 + c} is super-attractive, which means that the iterations of all the points around this cycle are attracted by it. in fact, all the points of the Julia's field for this value of c have iterations that are attracted by this cycle. N.B.: The equation (c^2 + c)^2 + c = 0 has four roots: 0; -1.76; -0.1+0.7i; -0.1-0.7i. If the calculations are done with
c = -0.1 - 0.7i, the result has the image of the rabbit by the symmetry whose axis is the real axis. This could be predicted since the value of c for this calculation is the conjugate of the parameter c previously considered. • 0 cyclic with an order of 4
In tha same way as Douady's, I determined values for c as solutions of ((c^2 + c)^2 + c)^2 + c = 0. I first chose c = 0.3 + 0.5i. In the same way as before, The super-attractors were plotted: 0, c, c^2 + c, (c^2 + c)^2 + c. Another root of the equation was computed: c = -0.16 + i. The results for this value of c are not very easy to comment. That must come from the increase of space for each iteration that is 0.01. I tryed to reduce the caluclation field and to increase the space step. But it tended to numerical and computer problems: No more memory available!!!