In the last post, we studied the dynamics of the logistic map , by looking at the plots of a few of its trajectories for various values of the bifurcation parameter . In this post, we try to get a few more insights into the nature of the dynamics, by making use of a versatile tool called the bifurcation diagram.
To construct a bifurcation diagram, we consider only the long-term dynamics. What do we mean by long-term dynamics? When we iterate a map such as the logistic map for a given value of , we start with some initial value . Let us suppose has been chosen such that the dynamics converges to a fixed point. Starting with , the map variable goes through several values before it ultimately converges to the fixed point. In such a case, the long-term dynamics takes only a single value – that of the fixed point. The several values assumed by before it converges to the fixed point, are ignored. Computationally, this is frequently done simply by leaving out the first few values, say the first 100 values , of . Similarly, if has been chosen such that the map dynamics settles into a 2-cycle, the long-term dynamics is just an oscillation between two values, and can again be computationally obtained by skipping the first few values of .
Next, to obtain the bifurcation diagram, we plot the value of , against the long-term values of . The sequence of long-term values is irrelevant; in a 4-cycle, we don’t care what is the order in which the four values occur in the dynamics… all that we plot is four points corresponding to those four values against the value of for which the 4-cycle occurs. This is a simple exercise can can easily be done on a computer with some graphics capabilities. It is very easily done using Matlab, Octave or similar numerical computation software. Here is a picture of the logistic map’s bifurcation diagram:
As can be seen from the bifurcation diagram, for , there is a single point on the bifurcation diagram, at 0. It indicates that the long-term dynamics is a fixed point, and the fixed point is at 0. For between 1 and 3, the map has a non-zero fixed point. Corresponding to , the bifurcation diagram has two points, indicating a 2-cycle. has four points on the bifurcation diagram, indicating a 4-cycle. Those who have a keen eye can possibly spot an 8-cycle too! The fully darkened areas of the diagram, pertain to those values of for which the long-term dynamics is neither convergent nor oscillatory, that is, the fully darkened areas indicate chaos.
The bifurcation diagram is an object of intricate structure and complexity. To end this post, let us have a look at a magnified version of the bifurcation diagram – from to .
Among other things, this magnified view clearly shows an 8-cycle. One can possibly see hints of a 16-cycle. In reality, the initial part of the bifurcation diagram contains an entire period-doubling cascade, consisting of all -cycles, . The diagram also shows a 6-cycle, a 5-cycle and a 3-cycle. Another interesting point that can immediately be noticed is that the darkened areas of chaos are interspersed with windows of periodicity – a phenomenon known as intermittency.
We shall have a more detailed look at the bifurcation diagram, in subsequent posts.