Iterated Logistic function with two initial conditions

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WHAT IS IT?

This is a model of an iterated logistic function f(x) = r * x * (1-x), which can be referred to as a logistic map, a nonlinear recurrence relation that plays a prominent role in chaos theory.

It is intended to visualize sensitive dependence on initial conditions (or the butterfly effect) as one of the key elements of systems with chaotic behavior.

HOW IT WORKS

In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.

To classify a dynamical system as chaotic, it must meet the following criteria:

It must be deterministic
Its orbits are bounded
Its orbits are aperiodic
It have sensitive dependence to initial conditions

This model focuses primarily on and deals with sensitive dependence on initial conditions (SDIC) which means:

• For any initial condition X1, there is another initial condition X2 near to it that eventually ends up far away • To predict the behavior of a system with SDIC requires knowing the initial conditions with impossible accuracy • Systems with SDIC are deterministic yet unpredictable in the long run

By inputting different intimal conditions (X1 and X2) and running the model one can observe that for very close initial conditions and the r-parameter=4 one can observe that these initial conditions after a number of iterations can end up far apart.

HOW TO USE IT

Input the X1 and X2 (initial conditions values)
Input r - parameter. It is recommended to select r close to 4 (which generates chaotic behavior of the system)
Select and input the number of iterations

Press the SETUP button, then press the GO button to run the model.

Observe the final X1 and X2 values shown by reporters and compare these values with the initial ones inputted before the model run.

Repeat the procedure with different X1 and X2 initial values and different number of iterations.

THINGS TO NOTICE

With the setting after the model is just opened two orbits that start very close together (X1=2.0 and X2=2.000001) end up (after 70 iterations) far apart (i.e. X1= 0.73844 and X2= 0.04755). This is known as sensitive dependence on initial conditions, or the butterfly effect.

You can notice that the plots for both initial conditions are identical until time-step 19. After this plots diverge dramatically. In this case step 19 is the limit for prediction. After this the behavior of the system cannot be predicted for these initial values.

THINGS TO TRY

Try different values for initial conditions, r-parameter and number of iterations. For any given initial conditions find out the limit for system predictability (i.e. # of time steps), after which the behavior cannot be predicted.

EXTENDING THE MODEL

The model can be extended by adding an additional plot for visualization of the difference between X1 and X2 with each time step.

RELATED MODELS

This is one of the models in a suit of models created to visualize some key concepts of Chaos Theory and Dynamical Systems. Most of the models are available on http://modelingcommons.org/account/models/2495

Of a special interest can be the models describing: * Iterated logistic function (with a final state-diagram) and * Bifurcation diagram for the logistic map

CREDITS AND REFERENCES

This simple abstract model was developed by Victor Iapascurta, MD. At time of development he was in the Department of Anesthesia and Intensive Care at University of Medicine and Pharmacy in Chisinau, Moldova / ICU at City Emergency Hospital in Chisinau. Please email any questions or comments to viapascurta@yahoo.com

The model was created in NetLogo 6.0.1, Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

This model was inspired by Introduction to Dynamical Systems and Chaos (Fall, 2017) MOOC by David Feldman @ Complexity Explorer (https://www.complexityexplorer.org/courses)

Comments and Questions

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Click to Run Model

globals [   ;; seting globals for two separate lists for
mylist-1    ;; two different initial conditions
mylist-2
]

to setup ;; setting two separate lists with their initial conditions
  ca
  set mylist-1 list (0) (X1)
  set mylist-2 list (0) (X2)
  reset-ticks
end 

to go                                         ;; procedure of iteration
  if r-parameter > 4 or X1 > 1 or X2 > 1 [    ;; with the results added to respective list
    ask patch 350 7
    [ set plabel "ERROR: R must be between 0 and 4.0 // X1 and X2 must be between 0 and 1.0."
      set plabel-color yellow ]
    stop
  ]
  set mylist-1 lput result-1 mylist-1 ;; bulding up lists as a result of iterations
  set mylist-2 lput result-2 mylist-2

  if ticks >= Num-iterations [ stop ] ;; stopping the model once
                                      ;; the inputted number of iterations is acheived
  tick
end 

to-report result-1                                         ;; procedure of iteration
  report  r-parameter * last mylist-1 * (1 - last mylist-1)
end 

to-report result-2
    report  r-parameter * last mylist-2 * (1 - last mylist-2)
end

There is only one version of this model, created almost 8 years ago by Victor Iapascurta.

Attached files

File	Type	Description	Last updated
Iterated Logistic function with two initial conditions.png	preview	Preview for 'Iterated Logistic function with two initial conditions'	almost 8 years ago, by Victor Iapascurta	Download

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