Cobweb plots

Cobweb plots are a way of visualizing iterations of a function.

cobweb plot for iterating cosine

For a function f and a starting point x, you plot (x, f(x)) as usual. Then since f(x) will be the next value of x, you convert it to an x by drawing a horizontal line from (x, f(x)) to (f(x), f(x)). In other words, you convert the previous y value to an x value by moving to where a horizontal line intersects the line y = x. Then you go up from the new x to f applied to the new x. The Python code below makes this all explicit.

Update: I made a couple changes after this post was first published. I added the dotted line y = x to the plots, and I changed the aspect ratio from the default to 1 to make the horizontal and vertical scales the same.

    import matplotlib.pyplot as plt
    from scipy import cos, linspace
    def cobweb(f, x0, N, a=0, b=1):
        # plot the function being iterated
        t = linspace(a, b, N)
        plt.plot(t, f(t), 'k')

        # plot the dotted line y = x
        plt.plot(t, t, "k:")
        # plot the iterates
        x, y = x0, f(x0)
        for _ in range(N):
            fy = f(y)        
            plt.plot([x, y], [y,  y], 'b', linewidth=1)
            plt.plot([y, y], [y, fy], 'b', linewidth=1)
            x, y = y, fy

The plot above was made by calling

    cobweb(cos, 1, 20)

to produce the cobweb plot for 20 iterations of cosine starting with x = 1. There’s one fixed point, and the cobweb plot spirals into that fixed point.

Next let’s look at several iterations of the logistic map f(x) = rx(1 − x) for differing values of r.

    # one fixed point
    cobweb(lambda x: 2.9*x*(1-x), 0.1, 100)

    # converging to two-point attractor
    cobweb(lambda x: 3.1*x*(1-x), 0.1, 100)

    # starting exactly on the attractor
    cobweb(lambda x: 3.1*x*(1-x), 0.558, 100)

    # in the chaotic region.
    cobweb(lambda x: 4.0*x*(1-x), 0.1, 100)

The logistic map also has one stable fixed point if r ≤ 3. In the plot below, r = 2.9.

cobweb plot for logistic map, r = 2.9

Next we set r = 3.1. We start at x = 0.1 and converge to the two attractor points.

cobweb plot for logistic map, r = 3.1

If we start exactly on one of the attractor points the cobweb plot is simply a square.

cobweb plot for logistic map, r = 3.1, starting on attractor point

Finally, when we set r = 4 we’re in the chaotic region.

cobweb plot for logistic map, r = 4

More posts on iterated functions

3 thoughts on “Cobweb plots

  1. The explanation would be made more clear if at least in one of those plots the “y = x” line was also plotted (perhaps with different style, different thickness, and/or different color).

  2. Hmm, it’s easy to graphically spot a fixed point (aka 1-cycle) with this plot, and I think you can tell if it’s stable by looking at the slope there, but it’s not obvious to me how to locate a 2-cycle.

    I’m thinking about moving a carpenter’s square along some fixed track, but can’t nail down the details. Any ideas?

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