International Encyclopedia of Systems and Cybernetics

K.DE GREENE warns: "… confusion can arise from regarding "cyclical" as synonymous with "periodic" (1988, p.283).

Periodicity supposes a clearly defined frequency, normally combined with an also well defined amplitude. Periodic curves are sine waves, ideally perfect ones. But perfect periodicity supposes the also perfect stability of environmental conditions, whether in astronomy or in economy.

Such a situation is merely an abstract scheme, which explains the general failure of purely formal forecasting.

In real situations we witness numerous combinations between cycles of very different frequencies and amplitudes.

Some cases are more or less manageable, when one of the superposed cycles is strongly dominant. These situations have precisely been those which gave rise to the general idea of cycle. Populations ecology, for example, offers such cases

However, this is exceptional. The normal case is the superposition of various or even numerous cycles of different frequencies and amplitudes, some of these quite comparable, others covering widely different time spans.

The composition of such a variety of characteristics yields normally curves that seem completely irregular, with unpredictable and sudden changes of orientations. (so-called chaos situations) A very good example is given by the stock market vagaries.

There are however some intermediate situations which look more promising:

1. When the respective frequencies are commensurable, even in the case when one period is so very much longer than the other or others that it may escape observation and analysis.

2. When the respective frequencies are incommensurable, but covering not excessive spans of time as to be still practically observable. In such cases, the complete cycle corresponds to the least common multiple of the frequencies. For example, a complex cycle of 1547 years would be the result of three interfering cycles of 7, 13 and 17 years.

However, the practical problem is quite intractable, because the detection of any regularities demands to begin with, a large series of observations of a generally confuse type of situations, covering a very long span of time.

Nevertheless, the concepts of periods embedment, of renormalization, of fractalization and of deterministic chaos may be of some help in the future.

→ Self-similarity in the WEIERSTRASS function.