INTRODUCTION TO CHAOS:
A SIMPLE CASE OF CHAOTIC DYNAMICS
D. C. Mikulecky
The simplest example of chaotic dynamics arises in the case of the logistic equation which has the following motivation. Imagine a growth process in ecology, economics, or some other related growth process. The system is monitored in discrete time steps, e. g. generations, years, etc. Let Xn represent the population size in the nth generation. The most naive growth law would be that the next generation depends on this one in a proportional way, with the growth factor, r, a being the constant of proportionality.
Xn+1 = rXn.
A somewhat more sophisticated model says that, if unchecked, the growth will follow an exponential law,
Xn+1 = rkXn.
If r > 1 , X tends towards infinity.
A more realistic model recognizes that growth is limited by limited resources, space,etc and that these impediments also increase with population. As Xn tends to get large, the growth rate tends to slow down. The simplest way to incorporate this is to let r = A(1-Xn). Then the expression is
Xn+1 = A(1-Xn)Xn.
With this model one can calculate the growth of this population on a hand calculator. If you try it, you must supply a numerical value of the growth parameter, A, and some initial population size, Xn (0 < Xn < 1).
Another way to make the calculation is by a graphical method. The figure below illustrates this method.
Figure 1: The "phase portrait" for the logistic equation in the stationary realm.
The equation, which describes a parabola, is plotted in figure 1 with the particular value for A being set at 2.5. This determines the exact shape of the parabola, which is of crucial importance. The line at 45 degrees is the locus of all points for which Xn+1 = Xn. By starting with the initial value on the horizontal axis (vertical line with arrow), the new value is found by drawing a verticle line to the parabola. [On your calculator this is equivalent to multiplying the initial value of Xn by (1 - Xn )A]. Project the point of intersection horizontally to the vertical axis to determine Xn+1. Then, to do the next iteration, extend this horizontal line until it intersects the 45o line. This converts Xn+1 into Xn for the next iteration. For this value of A (2.5), the process quickly converges on a particular stationary vale of X as can be seen on the graph as a set of ever diminishing partial rectangles. This final point represents the equilibrium for the system. [On your calculator, this value will always result in itself being calculated for Xn+1 when it is fed in as Xn.]
In graphical form to the left we see the successive values of Xi plotted by generation, i. After an initial transient phase, the system settles into a constant value in time.
The next example is for a slightly higher value of A and it results in periodic behavior.