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+--- Thread: [2014] The angle fractal. (/showthread.php?tid=930)
[2014] The angle fractal. - tommy1729 - 10/10/2014
If an initial value x converges to a fixpoint at an angle y , we can give that value pair(x,y) a corresponding color depending on the angle y.
We can make a plot that would then be the analogue of a fractal , the coloring based on the values y corresponding to the x's.
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That is the main idea ...
Some comments : the derivative at the fixpoints x_0 needs to be positive real.
SO :
Arg ( f ' (x_0) ) = 0
0 < Abs ( f ' (x_0) ) < 1
Also the case is simplest when f(z) only has 2 conjugate fixpoints.
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Therefore I seek :
Taking - without loss of generality ? - the fixpoints equal to +/- i .
g(z) = real entire = ??
f(z) :=
exp(g(z)) (z^2+1) + z
Arg ( f ' (i) ) = 0
0 < Abs ( f ' (i) ) < 1
And also f(z) ~ exp(z) z^A for some small real A.
Many solutions g(z) must exist , but which is best ?
And how do the " new fractals " look like ?
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Also can the angle be given by an integral ?
Contour integral ?
z_0 = initial value
z_n = f ( z_(n-1) )
lim n -> +oo
[z_n - z_(n-1)] / [ | z_n - z_(n-1) | ]
= contour integral ( z_0 ) ???
regards
tommy1729
RE: [2014] The angle fractal. - tommy1729 - 10/19/2014
So f(z) is of the form
f(z) =
exp(z + fake_ln( (- e^(-z) z^3 - e^(-z) z + 1 ) / (z^2+1) )) + z
, such that the derivative at both the fixpoints is a real Q.
If Q lies between 0 and 1 that can give a nice " angle fractal ".
However the case Q > 1 is also very intresting and gives an analogue of sexp for the superfunction of f(z).
This should be worth an investigation !
regards
tommy1729