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* BIFURCATION DIAGRAM OF LOGISTIC EQUATION MODEL *
By: Abdiel Emilio C‡ceres Gonz‡lez.

WHAT IS IT?

This model is an application of the famous Bifurcation diagram of logistic equation X[t+1] = k * X[t] * (1 - X[t]); with k=0..4 and X[t]=0..1

HOW IT WORKS

There are up to 10,000 turtles that has the job of paint one single point at X[t+1] on each iteration (In fact, only 400 turtles can be drawn, because the canvas is 400x200 pixels). The graphic represents each application of logistic equation over some value of the parameter k, remember that k is in [0,4] interval.

The first 40 iterates don't drawn any point, but you can see how is the dinamic behaviour.

HOW TO USE IT

SETUP button: Initialize the model, draw axes, optionally draw grid with 10 segments, and set up all turtles (SAMPLES in the model).

Go: kx(1-x) and GO buttons: run the model

SAMPLES slider: create <samples> number of turtles (values of k)

x0 slider: setup the X[0] value. This variable don't have effect, because initial conditions are not drawn.

kMin: setup the lowest limit of k

kMax: setup the greatest limit of k

rnd-color?: Boolean variable that its useful if you want to draw each point with a different color, to see (in long term) a vertical line of the same color.

grid?: Boolean variable that establish the grid with 10 horizontal and vertical divisions
The reporters
Time: the number of iterations
Min = kMin
Max = kMax
Half point = kMin + (kMax - kMin) / 2

The model works with 2 initial parameters: k and X[0].

Press SETUP, then Go [ or GO (1-step) for 1 iteration ]

THINGS TO NOTICE

- Set 20 samples to apreciate all points of graphic into each application
- Move kMin and kMax to see the windows of regularity (order between chaos).
- Turn on/off grid? variable to see (or not) the grid
- Turn on/off rnd-color? variable to see (or not) colors
- Cantor dust around k=4

THINGS TO TRY

- set 400 samples and 200 iterations (time = 200)
- set 1000 samples and 200 iteration
- set 10000 samples and 200 iteration. Observe and anotate the differences
- Move x0
- Where is the exactly value of k that one single fixed point change to periodic?

EXTENDING THE MODEL

You can write code for change vertical scale.

RELATED MODELS

The Lorenz system; Rabbits Grass Weeds model; Wolves and sheep; Logistic Growth.

CREDITS AND REFERENCES

To refer to this model in academic publications, please use:

Caceres-Gonzalez, Abdiel E. "Bifurcation Diagram of Logistic Equation Model". Universidad Ju‡rez Aut—noma de Tabasco, DACB. Complex Systems and Theoretical Computer Foundations Group. Cunduac‡n, Tabasco, MŽxico. June 2007. http://computacion.cs.cinvestav.mx/~acaceres/resources/software/NetLogoModels/LogisticEq/bifurcacion2.nlogo

CONTACT THE AUTHOR

Abdiel E. Caceres-Gonzalez,
Complex Systems and Theoretical Computer Foundations Group at
Universidad Juarez Autonoma de Tabasco,
Cunduacan, Tabasco, Mexico.
abdielc@acm.org,
http://www.cs.cinvestav.mx/~acaceres/
june 13, 2007.