different, ranging from 0 to 1, and a minor change in the initial value for x n will result in a totally different sequence of figures after just a few stages. Chaotic behavior is seen from a = 3.56994... onwards. If a graph is plotted we get a bifurcating tree that shows how periods double and then end in chaos (Fig. 1). Oddly enough the chaotic range shows stable islands with bifurcations that move in rhythms of three, six and twelve. Doubling periods leading to chaos are one typical route seen with non-linear equations for evolution dynamics, and the pattern is not limited to this kind of simple logistic equation. The chaos is predetermined, which is why it is also called a “deterministic” chaos. Eilenberger felt moved to paraphrase Hamlet, saying: “Though this be chaos, yet there is method in’t.” The logistic equation is the simplest kind of non-linear evolutional equation with feedback, using merely one parameter (a) and one variable (x n ). In reality, changes in rabbit populations are infinitely more complex. Other animal populations share their food resources and the weather also plays a role. The rate of reproduction is not constant, and the mortality rate is subject to diseases and natural enemies, i.e., it is also influenced by other animal populations, e.g., foxes, and the same holds true for their evolution. An epoch-making discovery was made by Mitchell Feigenbaum, a young physicist in the USA. He felt constrained by the thought structures of classical physics and looked for different approaches, among other things making an intense study of Goethe’s theory of color. The bifurcation diagrams suggested to him that higher laws may pertain to the route to chaos. He analyzed the diagrams and found that proportional laws applied. The points of bifurcation come closer and closer together just like a line of telegraph poles do with increasing distance when seen in perspective. The factor for calculating subsequent bifurcation points is a constant: d = 4.66920…, known as the Feigenbaum number. Surprisingly, the Feigenbaum number is an irrational number, a universal constant, rather like p. In France, Libchaber, also looking for new approaches in physics—he had studied Goethe’s scientific writings and particularly also Theodor Schwenk’s work on turbulent flow—examined the behavior of fluids on the way to turbulence. This also led to bifurcations and confirmation of the Feigenbaum number. The route to deterministic chaos could thus be defined. Once two bifurcations are known, it is possible to use the Feigenbaum number and determine all subsequent bifurcations up to the point of accumulation where chaos begins, and this does not require an equation or individual 58

Fig. 1. Population dynamics with feedback from growing conditions in the biotope. Abscissa: Control parameter “a” (rate of replication); ordinate: population. If the rate of replication is low, the population becomes extinct. If it is higher, a stable state develops, with the population increasing as “a” increases until a limit is reached. If “a” is borderline, bifurcation occurs, with the population periodically oscillating between two levels. A further increase results in further bifurcations with quasiperiodic oscillations of population levels. Finally the population shows chaotic oscillations (black area), though stable, ordered states (white stripes) occur at intervals. (After Cramer) calculations. Thus there is a universal law that governs the different mathematical equations and calculations. If you take two consecutive bifurcation elements and increase the smaller of the two by a factor of d = 4.66920 parallel to the vertical axis and d = 2.50290 parallel to the horizontal axis, at the same time rotating the graph so that top becomes bottom, the resulting images will be identical. 5 This potential for continually converting individual bifurcation elements into identical images indicates, furthermore, that the diagrams are self-similar. As early as 1904, the mathematician von Koch devised an absolutely regular but self-similar curve that was considered a mathematical monstrosity by other mathematicians until chaos theory began to develop. A Koch curve is produced from an equilateral triangle, with the points in the middle of each side displaced perpendicularly until they become aligned along a 59