The Logistic Equation

As seen in Chapter 1, the following equation is the general form of the Logistic Equation:

where 0<xsl,0<asl.

The Logistic Equation is a one-dimensional nonlinear feedback system. It is also a difference equation, as opposed to a continuous system, such as is obtained from partial differential equations. It is therefore a discrete system as well. As a difference equation, it lends itself to computer experiments with spreadsheets. One need only copy a formula down, in order to study its behavior.

We can create such a spreadsheet easily, using the following procedure:

fill,

1. In cell A1, place an initial value for the constant, a, between 0 and 1. Start with 0.50.

2. In cell B1, place an initial value for x of 0.10.

3. In cell B2, place the following formula:

Note that the value of a in cell A1 is treated as a constant.

4. Copy cell B2 down for at least 100 cells.

FIGURE 10.1 The Logistic Equation: convergence of x(t); a - 0.50.

FIGURE 10.1 The Logistic Equation: convergence of x(t); a - 0.50.

ITERATION NUMKR

ITERATION NUMKR

The Route to Chaoi 123

By plotting column B as a time series, we can study the transition of the system from stability to chaos.

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