VMI Chaos Lab, NetLogo Section


Notes

  • These programs make use of NetLogo 2.1 models.
  • Warning: it takes a couple of minutes to load an applet.
  • With all our NetLogo models, click "setup" first.

Experiments for 1D iterations, Lab 1

  • Graph of the logistic fuction with cobwebs
    • This model draws trajectories of the logistic function using the Koenig-Lemeray (KL) or cobweb method.
    • Lab 1. Graph of the logistic function (Load the applet into your browser) This is a model in NetLogo 4.1.3.
  • How to run the model
    1. Set the family parameter, R, with the slider
    2. Click "setup"
    3. Click "step" several times to follow the trajectory
    4. Click "clear" and then "step" to reveal the attractor
  • Suggestions
    • Study the attractor for increasing values of R
  • Tutorial video for Lab #1, 7 minutes

Experiments for 1D iterations, Lab 2

  • Response diagram of the logistic map family
    • The response diagram is a graphical portrait of the attractors of a scheme, that is, a family of maps. In this case, the logistic family is used.
    • Lab 2. Logistic Response (Load the applet into your browser)
  • How to run the model
    1. Click "setup"
    2. Click "make-it" to begin to draw the diagram (button turns black)
    3. Wait for about 30 seconds or so to complete the proces (button will turn blue again)
  • Suggestions
    • Begin with R-min = 0, and R-max = 4, for the full response diagram.
    • Choose other combinations of R-min and R-max to stretch out a thin vertical slice of the diagram.
    • Try to find periodic windows within the chaotic regime.
  • Tutorial video for Lab #2, 6 minutes

Experiments for 2D iterations

  • Iterations of the Dorband map family
  • How to run the model
    1. Click "setup"
    2. Click "place-particles" (the button turns black)
    3. Move cursor into the Graphic Window (large black square) and click to place a test particle (small green square)
    4. Repeat until you have a few well separated test particles
    5. Click "place-particles" again (the button turns blue again)
    6. Click "go" (the button turns black, the test particles swiftly jump along their trajectories leaving red dots, as each approaches its attractor)
    7. Click "go" again to stop the action (button turns blue again)
  • Suggestions
    1. The parameter "c" is set by the slider, which may vary from 0.0 to 1.0. Each setting of this slider determines one map of the Dorband map family
    2. Find the attractors of the Dorband map for various values of the parameter
    3. Hint: there is always either one or two attractors
    4. Fix a value of "c" using the slider
    5. Place some particles in initial positions in the graphic window, which shows the entire state space of the map, a 2D rectangle
    6. Press "go" and shortly after, press it again, to draw trajectories for each of the particles
    7. Use the "clear" button the erase the trajectories drawn so far, leaving the test particles (the small green squares reappear) in their current positions
    8. Continue until you think you have found all the attractors (red blotches)

Experiments for Basins of Attraction

  • Iterations of a quartic map
    • To be posted shortly
    • This applet iterates a map with two basins. The aim is to locate the basins by trial and error.
    • Lab 4. Basins discovery (Load the applet into your browser)
  • How to run the model
    • 1. setup clears and sets up everything
    • 2. place-particles, click the button, then click Graphics Window to place only one particle, click the button again to finish this step
    • 3. click the go button to begin to draw a trajectory, click it again when trajectory is long enough, to stop drawing
    • 4. change color of trajectory if you wish: change inkcolor (0.0 is black), retrace (can only be done once)
    • 5. if satisfied with the trajectory colors, cleanup (removes all turtles leaving trajectories)
    • 6. repeat steps 2-5 looking for new attractors and coloring basins
  • Suggestions

Revised 17 April 2009 by Ralph Abraham, <abraham@vismath.org>