globals [interval time sepG] turtles-own [xpos ypos] to-report scale-x [val] ;;a value between 0 and 4 report val * 9.8 end to-report scale-y [val] report val * 20 end to-report xSig [ kappa xAct ] report kappa * xAct * ( 1 - xAct ) end to draw-Axes ca create-turtles 1 set-default-shape turtles "minisquare" ask turtles [ set color black setxy 0 0 pen-down setxy max-pxcor 0 setxy max-pxcor max-pycor setxy 0 max-pycor setxy 0 0 die ] if grid? [ crt 10 ask turtles [ set color 7 set xcor (1 + who) * (39 / 10) pen-down set ycor max-pycor pen-up setxy 0 0 set ycor who * 2 pen-down set xcor max-pxcor pen-up die ] ] end to setup set time 0 set interval (kMax - kMin) / samples draw-Axes ask patches [set pcolor white] crt samples set interval (kMax - kMin) / samples ask turtles [ ifelse rnd-color? [set color random 256] [set color black] set xcor (1 + who) * (39 / samples) ;a value between 0 and 39 setxy xcor 0 set xpos kMin + ( ( who + 1 ) * interval) set ypos xSig xpos x0 ] end to draw-Point pen-down setxy xcor ycor + 0.001 setxy xcor ycor - 0.001 pen-up end to go ask turtles [ setxy xcor scale-y ypos if time > 40 [draw-Point] set ypos xSig xpos ypos ] set time time + 1 end @#$#@#$#@ GRAPHICS-WINDOW 223 10 953 419 -1 -1 18.0 1 10 1 1 1 0 1 1 1 0 39 0 20 CC-WINDOW 5 484 963 579 Command Center 0 BUTTON 7 51 167 84 Go: kx(1-x) go T 1 T OBSERVER T NIL SLIDER 7 129 215 162 samples samples 1 10000 500 1 1 NIL SLIDER 7 174 216 207 x0 x0 0 1 0.742 0.0010 1 NIL SLIDER 8 215 215 248 kMin kMin 0 4 2.8125 1.0E-4 1 NIL SLIDER 8 253 214 286 kMax kMax 0 4 4.0 1.0E-4 1 NIL BUTTON 7 10 165 43 Setup setup NIL 1 T OBSERVER T NIL MONITOR 223 421 273 470 Min kMin 3 1 MONITOR 904 420 954 469 Max kMax 3 1 MONITOR 563 419 644 468 Middle-point kMin + (kMax - kMin) / 2 3 1 BUTTON 67 89 167 122 Go (1 step) go NIL 1 T OBSERVER T NIL MONITOR 6 419 70 468 Time time 0 1 SWITCH 6 299 116 332 rnd-color? rnd-color? 1 1 -1000 SWITCH 118 299 214 332 grid? grid? 0 1 -1000 TEXTBOX 531 57 651 75 X[t+1]=k*X[t]*(1-X[t]) TEXTBOX 585 394 600 412 k TEXTBOX 232 199 257 217 X[t] @#$#@#$#@ * 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 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. CREDITS AND REFERENCES ---------------------- To refer to this model in academic publications, please use: Caceres-Gonzalez, Abdiel Emilio. (2007). 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. http://www.cs.cinvestav.mx/~acaceres/. Model available via email at abdielc@acm.org 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. @#$#@#$#@ default true 0 Polygon -7500403 true true 150 5 40 250 150 205 260 250 link true 0 Line -7500403 true 150 0 150 300 link direction true 0 Line -7500403 true 150 150 30 225 Line -7500403 true 150 150 270 225 airplane true 0 Polygon -7500403 true true 150 0 135 15 120 60 120 105 15 165 15 195 120 180 135 240 105 270 120 285 150 270 180 285 210 270 165 240 180 180 285 195 285 165 180 105 180 60 165 15 arrow true 0 Polygon -7500403 true true 150 0 0 150 105 150 105 293 195 293 195 150 300 150 box false 0 Polygon -7500403 true true 150 285 285 225 285 75 150 135 Polygon -7500403 true true 150 135 15 75 150 15 285 75 Polygon -7500403 true true 15 75 15 225 150 285 150 135 Line -16777216 false 150 285 150 135 Line -16777216 false 150 135 15 75 Line -16777216 false 150 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