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## Logistic Equation

We can use the ideas introduced in fixed-point iteration to investigate the logistic equation. The logistic equation is written as
 (1.17)

with the starting value satisfying and is a parameter. Choosing a value for , (1.17) is used repeatedly and the limit of as is obtained. Changing the value of the parameter , the procedure is repeated and the limiting value of obtained. Thus, the limiting value as a function of is built up.

The logistic equation can be used as a simple model of how a population changes in time. It illustrates some interesting ideas and produces features such as bifurcations and period doubling when the parameter is adjusted.

There are two fixed-points at and as can be verified by setting and in (1.17). To investigate whether these fixed-points are stable or not, we look at the convergence properties by calculating . Thus,

Now we have

and so the fixed-point at is unstable for .

For the fixed-point at we have

Thus, for (1.17) to converge to we need

and so
 (1.18)

Thus, for the logistic equation converges to the root at . That is, as . Note that for , and the error oscillates between positive and negative values without growing or decaying appreciably.

What happens for ? For just slightly larger than 2, numerical solutions to the logistic equation show that the scheme settles down and oscillates between two values. So instead of getting as we obtain

where and are the two different solutions. This is called a period two solution to the logistic equation and they exist for the parameter . The single solution is said to bifurcate when the parameter and this process is called bifurcation.

To obtain the period two solutions we apply the iteration scheme twice and assume that

as . Thus, we have

This is quite complicated but we know that and must be solutions (since for the original solutions) but they are unstable solutions. To proceed we note that the last term in the above equation can be expressed as

Hence, we have

Expanding the brackets and collecting powers of together we obtain
 (1.19)

This is of the form

To obtain the fixed points we set and so

The first two factors give the period one solutions and the square brackets factor gives the new period two solutions. Thus, they satisfy
 (1.20)

Solving the quadratic equation gives
 (1.21)

Notice that the square root only gives real solutions when and for we have . The period two solutions bifurcate from the solution.

As the parameter is increased we will eventaully reach a point at which the period two solutions will become unstable and the two solutions will each bifurcate giving a period four solution. The value of at which a bifurcation occurs is given by the condition

The calculation of is complicated. However, setting , where is given by (1.21), we obtain

Hence,

So period two solutions are stable for

For they become unstable and we get period four solutions. This process of bifurcations and period doubling occurs at particular values of , namely

until chaotic solutions (infinte period) arise.

Next: Summary Up: Other Iterative Schemes Previous: Summary
Prof. Alan Hood
2000-02-01