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# When Healing Becomes Educating, Vol. 2 - Waldorf Research Institute

When Healing Becomes Educating, Vol. 2 - Waldorf Research Institute

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eat of a butterfly’s wing may ultimately influence the whole meteorological system and cause a typhoon that will affect a whole continent. Like the butterfly, irrational numbers may produce incalculable changes in a complex system. The publication of Lorenz’s paper in a meteorological journal attracted little attention at first, and its full significance was only realized later. Then scientists working in all kinds of scientific disciplines in the USA, Europe, especially France, and the Soviet Union began to take up this work. In the USA, Robert May was teaching biology and looking at the dynamic changes in animal populations. He used the one-dimensional logistic equation x n + l = ax n (1 – x n ) Here, x n may represent this year’s population, x n+1 next year’s, and “a” a constant for the reproductive rate. We have to assume that an animal population, rabbits for example, cannot go beyond a certain number, as food resources would then be inadequate and the whole population might die out. This is accounted for by the product 1 – x n . The probability range for the population is between 0 and 1, and if this reaches 1 it will die out the following year. Population figures depend to a marked extent on the reproductive rate, and strangely enough, the system may develop in three different ways. Let us assume a rabbit population of x n = 0.1 reproducing at a rate of 2. This would give us x n + 1 = 2 x 0.1 (1 0.1) = 0.18. The following year we’d have x n + 1 = 2 x 0.18 (1 0.18) = 0.2952. The third year, x n + 1 would be 0.4161139, the fourth year 0.4859262, the fifth year 0.4996038, the sixth year 0.4999996. x n would thus be approximating 0.5, which would be a stable value for a rabbit population reproducing at a rate of 2. With the rate of reproduction and the available food stable, our rabbit population would automatically settle to a stable 0.5 for ever and ever. If the rate of reproduction were to rise, however, this would have a strange effect. If “a” is greater than 3, x n suddenly moves in two different directions. At a = 3.2, it would move towards 0.7994553 and 0.5130448, which would be the size of the rabbit population in alternate years. A further increase would cause a sudden further division at a = 3.44865..., with the period lengthening to four generations at 0.8749972, 0.3828195, 0.8269404 and 0.5008846. At a = 3.54431 we’d have 8 generations and at a = 3.56445 no less than 16. Periods would double at shorter and shorter intervals. So far, changes in our rabbit population have shown a strange kind of order. Further increases in rate of reproduction will cause this to come to a sudden end, with chaos intervening. Year by year the generation will be 57

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