Background
When we define functions recursively, such as
f(t) = f(t-1) * f(t-1) + c
A few cases might arise, as we keep increasing number of iterations, t.
Converge: Eventually, f(t) might converge to a zero or non-zero number.
Diverge: f(t) might keep getting bigger and bigger.
Periodic: f(t) might oscillate among 2 or more values.
Chaos: f(t) might produce random values.
There are cases where a function behave might differently with the choice of base case and other constants in this case f(0) and c. For,
Example: f(t) = r * f(t - 1) * (1 - f(t - 1))
f(0) = 0.2, r = 1, f(t) converges to zero.
f(0) = 0.2, r = 2, f(t) converges to 0.52
f(0) = 0.2, r = 3, f(t) is a attenuating-oscillating function
f(0) = 0.2, r = 4 f(t) is a chaotic function
For a function, for a specific set of points in a closed boundary used as the choice for f(0) or c, the f(t) might converge or diverge fast or slowly.
Mendelbrot and Julia set
For Mendelbrot set, the function is defined as the following
f(t) = f(t-1) * f(t-1) + c
f(0) = 0 and c is chosen from numbers from complex number domain from the following rectangular boundary,
bottom-left = -2.25 -1.5i
top-right = 0.75 + 1.5i
Algorithm to generate the plot
We count number of iterations, t, for which magnitude(f(t)) reaches to 4. If f(t) converges, we say it took MAX_ITERATIONS to converge. We put computed iteration counts in a 2D array. Plotting the 2D array in 2D plane gives a visualization of the Mendelbrot set. Yellow regions depicts those points that converges so for them iterations count = MAX_ITERATIONS. Darker points say that they diverge very quickly.
For Julia set, we fix c, and vary f(0) from a complex number boundary.
c = -0.35 + 0.65i and f(0) ranged from -2 to +2 in complex plane
When we define functions recursively, such as
f(t) = f(t-1) * f(t-1) + c
A few cases might arise, as we keep increasing number of iterations, t.
Converge: Eventually, f(t) might converge to a zero or non-zero number.
Diverge: f(t) might keep getting bigger and bigger.
Periodic: f(t) might oscillate among 2 or more values.
Chaos: f(t) might produce random values.
There are cases where a function behave might differently with the choice of base case and other constants in this case f(0) and c. For,
Example: f(t) = r * f(t - 1) * (1 - f(t - 1))
f(0) = 0.2, r = 1, f(t) converges to zero.
f(0) = 0.2, r = 2, f(t) converges to 0.52
f(0) = 0.2, r = 3, f(t) is a attenuating-oscillating function
f(0) = 0.2, r = 4 f(t) is a chaotic function
For a function, for a specific set of points in a closed boundary used as the choice for f(0) or c, the f(t) might converge or diverge fast or slowly.
Mendelbrot and Julia set
For Mendelbrot set, the function is defined as the following
f(t) = f(t-1) * f(t-1) + c
f(0) = 0 and c is chosen from numbers from complex number domain from the following rectangular boundary,
bottom-left = -2.25 -1.5i
top-right = 0.75 + 1.5i
Algorithm to generate the plot
We count number of iterations, t, for which magnitude(f(t)) reaches to 4. If f(t) converges, we say it took MAX_ITERATIONS to converge. We put computed iteration counts in a 2D array. Plotting the 2D array in 2D plane gives a visualization of the Mendelbrot set. Yellow regions depicts those points that converges so for them iterations count = MAX_ITERATIONS. Darker points say that they diverge very quickly.
f(0) = 0, c ranged from (-0.22 -0.7i to 0.75 -0.21i)
For Julia set, we fix c, and vary f(0) from a complex number boundary.
c = -0.77 + 0.13i and f(0) ranged from -1 to +1 in complex plane
Reference:
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