Teacher Information: Cob Web Plot

This is one of several manipulatives that illustrate "sensitive dependence on initial conditions," one of the defining characteristics of chaos and fractals.

There are basically two different shapes of curves (in blue), either a "Tent" or a "Logistic" curve (part of a parabola). Each is the graph of a function, f(x), in the unit square (0 to 1). Click to choose between the two curve shapes. The slider at the right determines the height of the graph, and with both kinds of functions, the black line represents the straight line y = x. The red dot is the initial point, the starting point. To adjust the starting point just click and drag the red dot; the initial value is shown below the graph.

The cobweb plot or "Verhulst diagram" is the path starting at the initial point, moving vertically until the point hits the blue graph, then horizontally right or left onto the y = x line, and then repeating the process (black line, vertically to blue graph, horizontally to black line, and repeat). For any given choice of blue graph and initial point, begin the cobweb plot by pressing the Play (top) button.

Students should be allowed to experiment freely with various cobweb plots by trying both graph shapes, changing the height with the slider just a little bit while keeping the same initial point, or changing the initial point for a fixed Tent or Logistic curve. When does the red dot end up at one end of the black line or the other? How closely can you make the path approximate a single square or rectangle? (Such paths would correspond to "attractors" in the chaos sense.)

For students with some experience working with functions, it is interesting to look at the cobweb plot in terms of repeating (or iterating) functions. For the x-value of the initial point, moving vertically to the graph changes the y-value to f(x). Then the path moves horizontally to the point on the y = x line with coordinates (f(x), f(x)). From there, moving vertically gives a y-value of f(f(x)), and so on, so that the successive x-values where the cobweb plot hits the y = x line are f(x), f(f(x)), f(f(f(x))),... What this manipulative shows is how small a change can make the sequence of functional-values behave completely differently, approaching 0 in one instance, approaching 1 in another, jumping almost back and forth among two or three values, or jumping chaotically all over the unit interval.