Logistic Growth Dynamics and Chaos

In the following paper we consider the so called Verhulst Dynamic(Logistic Equation) as an example of some characteristic aspects of Nonlinear Dynamics. The material is presented for German pupils of the final mathematics classes in the Sekundarstufe I (Gymnasium).

After considering simple relations between linear, exponential and logistic growth, we systematically inquire logistic time series and Feigenbaum- Bifurcations with the help of pocket calculators and personal computers. The essential characteristics of Chaos, "Merging, Periodicity and Sensitivity" are explained by means of the method of graphical iteration.

The method of graphical iteration, which is applied to linear systems, leads to the following important result:

Chaos inside linear systems is impossible, but it is not inherent in every nonlinear system. In other words: Nonlinearity is necessary but not sufficient for the existance of Chaos.

Furthermore discussed are questions of numerical error expansion and compression on a more intuitive-geometrical level. Graphic worksheets are used to support these questions.

The last part consists of references to the phenomena of sensitivity and bifurcation as well as to their consequences for dynamic systems in biology, ecology, climatology and other systems.

Finally a sequence for eighteen/ nineteen teaching units(45 minutes) is presented which can be shortend at the teacher’s discretion.

It intends to explain some central aspects of nonlinear dynamics and chaos to pupils of the Sekundarstufe I with the help of numerical experiments and simple graphical methods. Predominantly intuitive and illustrative method should be employed instead of difficult mathematics. In order to achieve this aim the use of worksheets which enable pupils to work independently is of major importance.

Note:

This article summarizes a paper comprising 74 pages which can be found in the following Part E.

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