Parameter perturbation and exotic "bifurcation" fractals

[bkercso]

Recently I tried sinusoidal perturbation on the parameter of systems which produces bifurcation fractals, like the logistic equation. I also experimented with swap the stepped parameter with an other property, like amplitude or frequency of the perturbation and generate 2D images of that.
Doesn't high mathematics follow, only high res images and some code.

These fractals were generated with Bifurcation Fractal Plotter:
http://www.fractalforums.com/windows-fractal-software/bifurcation-fractal-plotter-biffrapl/

Some fractals cannot be perturbed: fractal -5, 13, 14 and 15.

Example: parameter perturbation on the logistic map:

Code:

x:=[parameter+amplitude*sin(iterationNumber*frequency)]*x*(1-x);

Img #1: Parameter perturbed logistic map, Ampl=0.1, Freq=1E-5


Img #2: Parameter perturbed logistic map, Ampl=Freq=0.1


Img #3: Parameter perturbed logistic map, Ampl=0.1, Freq=1


Img #4: Parameter perturbed logistic map, Ampl=0.1, Freq=10


Img #5: Parameter perturbed delayed logistic map (fractal -1 in my program), Ampl=0.1, Freq=1E-4. Its something buddhabrot-like...


Img #5a: Parameter perturbed delayed logistic map, Ampl=0.01, Freq=1E-3

[bkercso]

Img #6: Parameter perturbed coupled logistic map, Ampl=0.1, Freq=1E-4; only the plotted system is perturbed


Img #7: Parameter perturbed coupled logistic map, Ampl=0.1, Freq=1E-4,; with perturbation on both systems, we get a more boring image:

[bkercso]

Img #8: A zoom of Img #2
Set:

Zoom:


Img #9: Zoom of Imtg #8
Set:

Zoom:


Img #10: Zoom of Imtg #8
Set:

Zoom: if we do more iterations, this microstructure goes to subpixel size - just see in Img #9, on left

[bkercso]

Img #11: Zoom of Img #2
Set:

Zoom:


Img #12: Zoom of Img #6
Set:

Zoom:

[bkercso]

Now some "exotic" bifurcation fractals follow! Exotic, because perturbation amplitude or frequency is stepped, instead of the parameter.

Img #13: Delayed logistic map (fractal -1), Parameter=2.2, Ampl=-1E-4..7.2E-2 (stepped), Freq=0.1


Img #14: Delayed logistic map (fractal -1), Parameter=2.25, Ampl=0.01, Freq=1E-3..10 (stepped)


Img #15: Delayed logistic map (fractal -1), Parameter=2.25, Ampl=0.1, Freq=1E-3..10: the behaviour is divergent at many frequencies (blank areas)

[matsoljare]

I can imagine many of them would produce interesting sounds, but i'm not sure how to go about with it...

[bkercso]

Equations are on this topic:
http://www.fractalforums.com/new-theories-and-research/bifurcations-fractals-discovery/

I perturb the parameter r:

Code:

r:=r_original+Ampl*sin(ActualIterationNumber*Freq)

[bkercso]

Img #16: Similar as Img #13, but with Freq=1. Calculated as a single attractor fractal, without any trials to change attractor


Img #17: Similar as Img #16, but with Freq=10. Calculated as a single attractor fractal, without any trials to change attractor. The break in the lines on the right is not an error: this is a multiple attractor fractal and that break is a spontaneous attractor change!


Img #18: Similar as Img #17, but calculated as a multiple attractor fractal, with 1000 trials to change attractor and 3000 avg. iter/pix. The "noise" on the left is "first few iterations", because I maximised the noise filtering iterations as Maxiter/2. As this noise filtering (eliminating first few iterations) is required after each trials to change attractor, there were too few rejected iterations here.