Do
not click on or manipulate the figure at the right; it is only an image. The
interactive interface is below.Pattern Perception in a Dynamic Universe
Exemplar 1 Applet
Fundamental Frequencies:
Attractor and Sub-Attractor Patterns
CITATION: Malloy, T.E., Butner, J., & Jensen, G. C. (In Press). The emergence of dynamic form through phase relations in dynamic systems. Nonlinear Dynamics, Psychology, and Life Sciences.
How might the processes which enable apparent motion be a basis for pattern perception in general?
Instructions
Information about Exemplar 2
Interactive Simulation: Fundamental Frequences of Attractor Cycles
Do
not click on or manipulate the figure at the right; it is only an image. The
interactive interface is below.
Instructions Top
Adjusting your Computer's speed to your Monitor's speed: Most monitors cannot paint accurately faster than 66 to 77 frames per second (fps).
Moreover, apparent motion effects are most robust when stimuli are presented about 25 frames per second (fps).
Use Delay. Before (or after) you push the Play button click the Use Delay radio button (highlighted in yellow) and then set the Delay (between iterations) slider (highlighted in yellow) so that the fps indicator (highlighted in violet) falls somewhere between 25 and 66 fps. In that range you should experience apparent motion effects. If your computer is slow, you may not need to use the Speed control.
Clicking on the Delay Slider Bar. To get finer-grain adjustments, you may click on the slider bar for the Delay Control, when you click on the delay bar the delay will increment (or decrement) a small amount.
Adjusting Window Size. The critical manipulation you will do is adjust the window size. This adjustment sets the phase relations among (1) your perceptual ability to perceive apparent motion/stability, (2) the dynamics of the system, and (3) the representation of those dynamics in a window.
Highlighted in blue on the illustration is the slider bar for adjusting theWindow Size variable. Just drag the slider along the slider bar to increase or decrease the Window Size.
Clicking on the Window Size Slider Bar. To get finer-grain adjustments, you may click on the slider bar for the Window Size Control, and the Window Size will increment (or decrement) one iteration.
Separator Bar. You may want to seem more iterations than the output frame (the part of the applet where you see the black and white squares of the historical trace) shows. You may drag the separator bar (highlighed in orange) to increase the size of the output frame. Or, you may click the little left-pointing arrow (highlighted by an orange oval toward the top of the illustration) to expand the output frame to its maximum size and to hide the control frame. If you do, remember you can click on the little right-pointing arrow to return the output frame to its normal size and to show the control frame.
Perturbing the System. Although not particularly related to the theoretical points we are making here, you may want to perturb the system and examine other basins. The perturb button will do this. The slider bar associated with the Perturb button adjusts the number of nodes whose states are reversed when you click the perturb button. Do not perturb the system until you finish the set of experiments outlined below.
Basic Information about this Dynamic System
We will repeat the figures from the NDPLS journal article for convenience here
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| Figure 1. Four highlighted repetitions in the historical trace of an attractor of length L=18 found in this a small system. The system's nodes are the vertical axis and the iterations (time) are the horizontal axis. |
Attractor Cycle Length = 18. This small dynamic system (as first loaded before you perturb it) will have an attractor cycle length, L=18. We call this its fundamental frequency. Some integer multiples of 18 are 18, 36, 54, 72, 90, ... When you look at the interactive interface, the output should similar to Figure 2
Nodes with attractor Sub-cycle lengths 1, 2, and 3 are also present in the dynamics. First find find sub-cycle of length 2 (See Figures below). (Sometimes we refer to sub-cycles as sub-basins which is not technically correct.)
Finding Fundamental Frequencies of Attractors
Click "Use Delay" This will allow you to adjust speed.
Press Play. First: Adjust the speed to between 25 and 66 frames per second as instructed above.The output panel of the interface (left) should show what appears to be a frozen pattern of black and white squares that looks identical (other than the highlighting) to Figure 1, above. In fact is not static but is being painted (on the interface below) at many frames per second (fps). So you are seeing apparent stability.
Explore. Play with Window Size (which is the number of iterations being painted to the screen at any moment).
Find Cycle Length, L =18. Notice that multiples of 18 produce apparent stability in the dynamics.
Finding Sub-cycles of attractors in basins
Find Sub-cycle Length = 2. Move the slider bar for the Window Size (in iterations) to 70. You will notice that the historical trace of all nodes that have SubL=2 will freeze (apparent stability). Any Window Size that is an integer multiple of 2 will reveal the nodes that have sub-basins of length 2.
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| Figure 2: Sub-basins (sub-cycles) of length 2. All nodes (rows) have been dimmed out except those rows that have sub-basins of SubL=2. For the overall basin, the length L=18. The system's nodes are the vertical axis and the iterations (time) are the horizontal axis. |
Find Sub-cycles Length = 3. Move the slider bar for the Window Size (in iterations) to 69. You will notice that the historical trace of all nodes that have SubL=3 will freeze (apparent stability). Any Window Size that is an integer multiple of 3 will reveal the nodes that have sub-basins of length 3.
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| Figure 3: Sub-cycles of length 3. All nodes (rows) have been dimmed out except those rows that have sub-basins of SubL=3. For the overall basin, the length L=18. The system's nodes are the vertical axis and the iterations (time) are the horizontal axis. |
After you play with the specific basin that loaded up, you can press the Pertub button which (most likely) will shift the system into other basins. In this system, all attractors have a fundamental cycle length of L=18. But difference basins have attractors that have differences in which nodes have sub-cycles and which nodes do not.
More can be found at www.psych.utah.edu/dysys.