JULIA SET GENERATOR
In the real world, it is impossible to calculate infinitely many iterations for the confident result whether the orbit diverges or not. In practice, there exists the term Threshold radius $r(c)$ defined by the equation $$r(c) = max(2,|c|).$$ It is proven that if the magnitude of any $z_{k}$ in the orbit exceeds the $r(c)$ then the orbit diverges and thus its initial value is not in the Julia set. Moreover for the Mandelbrot set only $2$ is considered without counting $max(2,|c|)$.
Converging
They converge to their fixed point. This type is located in the cardioid (heart-shaped part) of the Mandelbrot set.Diverging
They diverge to infinity. This type is located outside the Mandelbrot set.Periodic
The values of the orbit are repeated in the same order over and over again. There exists a special case of periodic orbit so-called preperiodic orbit, where the orbit becomes periodic after the first few values, which are not in the periodic cycle. Periodic orbits are located in the period-$n$ bulbs, where $n$ depends on the period of the orbit.
On the boundary of the cardioid are attached infinitely many disks usually called bulbs and then other bulbs are again attached to them etc. Because the critical orbit of $c$
from any bulb is periodic, the name bulb is specified by the period of this orbit as the period-$n$ bulb.
This part of the application mainly focuses on the interesting fact that the critical orbit of $c$ chosen from
some part of the Mandelbrot set corresponds to the period of the orbit of the initial value chosen inside the Julia set
generated for the same $c$.
By moving the cursor on the upper (bottom) screen the exact coordinates of the cursor are shown in the box in the left bottom (upper) corner of a panel and then in the middle of the bottom panel
application returns the behaviour of an orbit with the initial value chosen by the cursor.
After the click on any location in the upper screen, the critical orbit with the initial value $z_{0} = 0$ is drawn for the $c$ (equals to $z_{1}$) selected by cursor position.
Then the Reset orbit button clears all rendered orbits.
Selected $c$ is also automatically saved to the $c$ coordinates in the bottom screen for generating the Julia set using the Generate button.
Because of that property, there is a possibility of comparing the Mandelbrot set and the Julia sets orbits for the same $c$.
If the users do not want to use the pre-saved value $c$, they can enter any value manually.