Mouse Map

1 collaborator

WHAT IS IT?

NB: For better model performance/visualization it is recommended to download the program and run it on your PC, because on the server it is very slow.

This model is an example of the bifurcation diagram of the Gauss iterated map.

The β parameter is shown on the horizontal axis of the plot generated in NetLogo World View Window and the vertical axis shows the set of values of the iterated Gaussian function f(x) = exp (-α * X^2) + β .

The map is also called the mouse map because its bifurcation diagram for certain α-values resembles a mouse.

For certain α-values the map exhibits chaotic behavior after a period doubling cascade.

HOW IT WORKS

Every time-step (tick) an iteration is performed and as a result a point is generated (i.e. a green color turtle is created) and it is plotted on the plane (where Y-coordinates are obtained as a result of Gaussian function iterations and X-coordinates represent every tick β-increments of 0.00008, with every new β-value participating in the next-step iteration by definition).

The preset value for α is 4.0. The α-value can be modified by the respective slider in the interval [3; 10].

HOW TO USE IT

Gauss map is a nonlinear iterated map of the real numbers into a real interval given by the Gaussian function:

f(x) = exp (-α * X^2) + β

where α and β are real parameters.

The range of β-parameter in this model is between '-1.0' and '1.0'

For certain α-values bifurcations are followed by chaotic behavior:

First bifurcation happens with β = approximately -0.75 (α = 4.0)
With α = approximately 4.7 the bifurcation cascade is followed by a zone of chaotic behavior, and chaos become more evident with α over 4.9 (and β > -0.5)

During the model run one can observe the result of iterations in time (by ticks): new turtles/points being created and plotted.

THINGS TO TRY

At first, run the model with preset α = 4.0. Notice how iterations generate a 'mouse shape' bifurcation diagram.

Play with the values of α using respective slider. Notice how α-values influence the shape of bifurcation diagram.

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 model describing the bifurcation map for iterated logistic function.

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 [     ;; setting the globals for the model functioning
  mylist-x
  mylist-r
  num-iterations
  axis
 ]

breed [ m-turtles m-turtle ]    ;; these are turtles to be ploted on
                                ;; on bifurcation diagram

to setup   ;; general setup procedure
  ca
  set mylist-x  list (0) (-1)     ;; setting the list for turtles x-coordinates
  set mylist-r list  (0) (-1)    ;; setting the list for turtles y-coordinates
  create-axis                   ;; creating coordinates axes
  setup-c-origin                ;; and their components.
  setup-coordinates
  set num-iterations 22500    ;; this is the max number of iterates which
                              ;; "keeps" the results of iterations in the
                              ;; limits of the model world
  reset-ticks
end 

to create-axis
  ask patches with [ pxcor = (- 200) ] [ set pcolor white ]
  ask patches with [ pycor = (- 100) ] [ set pcolor white ]
end 

to setup-c-origin

ask patch -185 140
  [ set plabel "f (x)"
    set plabel-color yellow
    ]
  ask patch 190 -105
  [ set plabel "beta-parameter"
    set plabel-color yellow
   ]
end 

to setup-coordinates
ask patch -185 -95
[ set plabel "-1.0"
set plabel-color white
  ]

 ask patch -100 -95
[ set plabel "-0.5"
set plabel-color white
  ]

 ask patch 0 -95
[ set plabel "0"
set plabel-color white
  ]

  ask patch 100 -95
[ set plabel "0.5"
set plabel-color white
  ]

  ask patch 200 -95
[ set plabel "1.0"
set plabel-color white
  ]

  ask patch -185 155
[ set plabel "1.5"
set plabel-color white
  ]

  ask patch -185 100
[ set plabel "1.0"
set plabel-color white
  ]

  ask patch -185 50
[ set plabel "0.5"
set plabel-color white
  ]

   ask patch -190 0
[ set plabel "0"
set plabel-color white
  ]

   ask patch -185 -50
[ set plabel "-0.5"
set plabel-color white
  ]
end 

to go                                        ;; procedure of iteration

  set mylist-x lput result mylist-x               ;; generation of the list with
  repeat 30 [set mylist-x lput result mylist-x]   ;; x-coordinates of turtles

  set mylist-r lput result-r mylist-r         ;; generation of the list with
                                              ;; x-coordinates of turtles
  if ticks >= Num-iterations [ stop ]    ;; condition for stopping the model once
                                         ;; the preset number of iterations is reached
  create-m-turtles 1 [        ;; creating a turtle every iteration/tick
    set color green           ;; and ploting it according to the coordinates
    set shape "circle"        ;; as values in the respective lists
    set size 0.5
    set xcor (last mylist-r) * 200
    set ycor (last mylist-x) * 100
    ]
  tick
end 

to-report result                                               ;; reports the result of iterations
  report  exp ((- alpha-parameter) * last mylist-x * last mylist-x) + last mylist-r
end 

to-report result-r                  ;; reports the result of r-increments with every iteration/tick
  report last mylist-r + 0.00008
end

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

Attached files

File	Type	Description	Last updated
Mouse Map.png	preview	Preview for 'Mouse Map'	almost 8 years ago, by Victor Iapascurta	Download

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