So, let's review a bit. A simple iterative rule can lead to stability or fluctuation or chaos, depending on what value one chooses for a parameter of the iteration (in the present case, k, the steepness of the inverted curve relating current to next population size). The figure below gives you a summary of that, one you can play with not only to review but also to discover new things on your own.
The left part of the figure is the staircase sort of diagram you've seen before. The right part of the figure is something new, a "bifurcation diagram", which helps to visualize changes in the behavior of the system as the value of k varies. The parameter k is plotted along the x-axis of the bifurcation diagram (from 1 on the right to 4 on the left). What the curve shows is the population behavior for every k value when starting at a particular initial population size (x). For small and medium k values (to the left) what the bifurcation diagram shows is that (as we found earlier), the population reaches a single, stable population size (which increases with increasing values of k). Above k=3, the plot "bifurcates" (hence the name of the diagram) into two branches. This is the region where, as we saw earlier, the system oscillates between two population sizes. For still greater k, there are additional bifurcations, so that at k=3.5, for example, the population cycles stablely among four values (you can see this stable four cycle in the staircase diagram to the left). For even larger k, the plot blurs into many points for most k values. This is the chaotic region which we described earlier.
The bifurcation diagram is constructed by taking a particular value for the starting population size (x=0.125 is the default value, as shown below the staircase diagram) and then, for each value of k, iterating it a number of times (70 is the default value, as shown in the left slider above the diagram). To make it more likely that the values displayed will be those which occur after the system has achieved stable behavior, values for the initial iterations are discarded (20 is the default value, as shown in right slider above the diagram), and only the values for later iterations are plotted in the bifurcation diagram.
You can look at the staircase construction for any given k (and hence see what leads to the different parts of the bifurcation diagram) by clicking at an appropriate location below the bifurcation diagram (the chosen k value is displayed below the diagram). For some k values (particularly smaller ones) you may need to reduce the number of skipped iterations to see the whole staircase construction. For others you may want to increase both the skipped iterations and the total number of iterations to be sure the constructed patterns are stable. You may also want to zoom in on parts of the staircase construction by clicking and dragging across interesting regions (click once on the construction to return it to its original scale).
The figure above also gives you an easy way to determine whether the general behavior of the system for a particular k depends on the starting population size. You can change the initial population size (without changing k) by clicking on any point below the staircase diagram. Try it. You'll find that the staircase diagram may change in detail, but generally stays pretty much the same (though for some x values it may take longer for the values to stablize, so you'll have to increase the number of iterations. Notice anything else? We'll come to that.
When you selected new initial population sizes, it was not only the staircase diagram but also the bifurcation diagram that changed somewhat but stayed generally pretty much the same. That's particularly interesting, since the bifurcation diagram itself is an interesting (at least), attractive (at least somewhat), and (at least fairly) orderly entity, which you have just discovered itself has some stability, and hence reality. To put it differently, by using computers to explore iterative systems, we have actually created a new kind of order that had never been seen before, one that exists only as a result of iteration.
Here's the bifurcation diagram again, in a slightly different form, which we can use to further explore this newly discovered order.
Remember the first bifurcation, where increasing the k value shifts the population behavior from one stable point to an oscillation between two? And the second bifurcation, yielding an oscillation among four stable values? What's happening at a larger scale (the first bifurcation) is being repeated at a smaller scale (the second bifurcation). One might guess that this keeps happening. You can test this prediction yourself by clicking and dragging across the region of the second bifurcation to zoom in on it. And indeed you'll find that it too leads to a bifurcation. And you can zoom in on that (the same way) and ... ? In practice you can do this only a few times, because of the limitations of our computer and yours, but in principle ... ?
Systems like this, that look the same at a variety of different scales, are called "fractals", and this is another aspect of the kind of order which the rapid iteration made possible by computers has made it possible to create and explore. No one as yet fully understands how much order exists in patterns of this kind, nor even whether there is some way to ask the question which would reveal all the forms of existing order. So you can explore the question yourself, by zooming in on various other parts of the bifurcation diagram and seeing whether you can find additional orderly features.Serendip