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Author Topic: Bifurcation fractals discovery  (Read 13649 times)
Description: Hi res. images and equations of Logistic map etc.
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bkercso
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« on: October 18, 2012, 05:12:36 PM »

Hi!

I can find a lot of beauty images about several fractals on the net, but very few about bifurcation fractal(s) and those are low quality ones.
Let this topic about this types of fractals, which are also very nice ones I think.
I used linear coloring with saturation (white color) and logarithmic x-axis, except if I write other.

(I use my self developed program for fractal generating, it was written in Pascal. It quite simple, a little hard to set zoom area and generating image with this, but I like it. smiley )

First: The logistic map: x(i+1)=r*x(i)*(1-x(i)) ; where 0< x(1) <1 , 0< r <4
(http://en.wikipedia.org/wiki/Logistic_map)

the red lines show the first few iterations


the first few iterations, log. coloring


x(i+1)=r*x(i)*(1-x(i)^2) ; where -1.2< x(1) <1.2 , 1< r <3
zoom


if x(i)<0.5: x(i+1)=r*x(i), else: x(i+1)=r*(1-x(i)) ; where 0< x(1) <1 , 1< r <2
log coloring

« Last Edit: May 25, 2015, 01:06:09 AM by bkercso » Logged
bkercso
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« Reply #1 on: October 18, 2012, 05:46:12 PM »

Delayed logistic map: x(i+2)=r*x(i+1)*(1-x(i)) ; where 0< x(1), x(2) <1 , 1.99< r <2.271


log(log(x-axis))


zoom, log(log(x-axis))


zoom


zoom, log x-axis (?)

« Last Edit: May 27, 2015, 03:16:42 PM by bkercso » Logged
bkercso
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« Reply #2 on: October 18, 2012, 05:47:09 PM »

Delayed, averaged logistic map: x_temp=(x(i+1)+x(i))/2   x(i+2)=r*x_temp*(1-x_temp) ; where 0.83< x(1), x(2) <1.165 , 3.8< r <4.617


zoom


zoom


zoom, log coloring,


zoom


zoom, log coloring


zoom

« Last Edit: May 27, 2015, 03:20:44 PM by bkercso » Logged
bkercso
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« Reply #3 on: October 18, 2012, 05:55:05 PM »

Delayed, geometric averaged logistic map: x_temp=sqrt(abs( x(i+1)*x(i) ))   x(i+2)=r*x_temp*(1-x_temp) ; where -1.215< x(1), x(2) <1.22 , 3.68< r <4.828


with different saturation level in coloring


first few iterations, log coloring


zoom, first few iterations, log coloring


zoom, first few iterations, log coloring
« Last Edit: May 28, 2015, 12:41:24 AM by bkercso » Logged
bkercso
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« Reply #4 on: October 19, 2012, 12:53:43 AM »

The next one is the phase space of a simulated physical system: a dripping tap.
The model was simple:
The water drop was a mass point, which sits the end of a spring. The spring force corresponds to the force from the surface tension of water. When the elongation of the spring was bigger than a constant, the drop gets smaller (dripping). When dripping, the mass reduction of the drop and the reduction of elongation are depend on the actual velocity of the drop.

Note that this simulation do not gives you a point on the picture at every iteration, only if one drip is dropped. So calculation time is much longer than in the previous cases.



The bifurcations are not smooth, maybe bacause of the suddenly changes of the elongation of spring when drop. I didn't calculate with the oscillations of the spring... Edit: Or I should use smaller dt, or 4th order Runge-Kutta method instead of Euler one (see this later).
X-axis: parameter: flow rate of water
Y-axis: velocity of drops, 200,000 point/column (dt=5E-6 sec)
« Last Edit: May 28, 2015, 12:50:18 AM by bkercso » Logged
bkercso
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« Reply #5 on: October 22, 2012, 10:00:44 PM »

The next is the Bifurcation diagram of double pendulum
(Description of the double pendulum: http://en.wikipedia.org/wiki/Double_pendulum
Equations I calculated with are here on page 13 (optimize these before run!): http://online.redwoods.cc.ca.us/instruct/darnold/deproj/sp08/jaltic/presentation.pdf)


Details of my setup (in SI system): L1=L2=1, m1=m2=1, theta1_initial=0, theta2_initial=0..pi, g=9.81, dt=1E-5, 3000 points/column
X-axis: theta2_initial (log. axis)
Y-axis: values of theta2 when m1 stops (w1=dtheta1/dt=0)
Calculation time: 17 hours @2.5GHz


Zoom (dt=5E-6), log coloring (As you can see the pics are not symmetric...)


I'm working on higher quality images... smiley
« Last Edit: May 27, 2015, 03:28:03 PM by bkercso » Logged
kram1032
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« Reply #6 on: October 22, 2012, 11:17:18 PM »

there are some awesome works in this.
For some reason I never thought of this before, but what about a magnetic double pendulum?
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bkercso
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« Reply #7 on: October 23, 2012, 03:01:13 AM »

Thank you.
Good question. Maybe this helps: http://www.math.tamu.edu/~mpilant/math614/StudentFinalProjects/Musick_Final.pdf  cheesy
But if you use real magnetic field model, not uniform, the system difficulty get more-more higher. Easier to do finite element simulation for these systems than calculate exact equations I think.
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cKleinhuis
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« Reply #8 on: October 23, 2012, 04:38:48 AM »

that double pendulum seems to be awfully nice chaotic smiley
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KRAFTWERK
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« Reply #9 on: October 23, 2012, 08:35:18 AM »

that double pendulum seems to be awfully nice chaotic smiley

Yes! I love the shapes of the left part of this image:
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Alef
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« Reply #10 on: October 23, 2012, 05:16:34 PM »

Nice pics. Bifurcation maps hadn't been meant to be pretty, they mostly are used to show that the mandelbrot set is meaningfull;)
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bkercso
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« Reply #11 on: October 23, 2012, 06:37:34 PM »

What would be a delayed Mandelbrot set?...
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fractower
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« Reply #12 on: October 23, 2012, 08:28:21 PM »

It might be interesting to consider the complex maps.

Z(i+1) = C * Z(i) * ( 1 - Z(i)) where C is a complex version of r.

I suspect there will be restrictions on the phase of C for bounded maps.
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bib
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« Reply #13 on: October 23, 2012, 09:02:56 PM »

Very interesting topic! For some reason I simply never asked myself what would a bifurcation diagram look like with another function than the logistic map. Looking forward to the next images.
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bkercso
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« Reply #14 on: October 23, 2012, 10:29:41 PM »

Quote
It might be interesting to consider the complex maps.
Z(i+1) = C * Z(i) * ( 1 - Z(i)) where C is a complex version of r.
I suspect there will be restrictions on the phase of C for bounded maps.
This gives a 4D fractal: C(re), C(im), Z(re), Z(im). Interesting...
« Last Edit: October 23, 2012, 10:37:51 PM by bkercso » Logged
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