Brief attempt about chaos and beauty – Self-similarity

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The biologist Robert May took the next significant step in chaos research in the early 1970s. He was concerned with the prediction of biological populations – a field in which the results also depend very much on the initial conditions. If limiting factors like predators and limited food resources were not taken into account, the process would be straightforward, with the outcome of one year serving as the baseline for the next. The simplest equation that includes limiting factors is:

Next year's population = 
Birth rate * Current population * (1-current population)

Robert May played with different birth rates to see what would happen to the results. He realized that the population size settled to a stable value when the birth rate was low and also increased as the birth rate increased. This behavior held true up to a certain point. If the birth rate passed it, the graph would split, and the result would jump back and forth between two values: A and B. For one year, the result would be A, and for the next year, it would be B, and this cycle would continue indefinitely. If he calculated with an even higher birth rate, there were four cyclically changing results, and these doublings (bifurcations) came faster and faster as the initial value continued to rise. However, once the population reached a certain size, it became impossible to predict the outcome. This was the limit beyond which chaos set in. However, this was not the most intriguing aspect of the graph. What was more intriguing was that the graph showed white stripes in this chaotic region, where the equation repeated the doublings before falling back into chaos. So, deep inside, the graph shows exact reduced copies of itself. We refer to this behavior as self-similarity, which is a crucial feature of chaotic systems.

While May acknowledged the geometric peculiarity of his diagram, he viewed it as a mathematical oddity from a biological perspective. It was left to a physicist, Mitchell Feigenbaum, to recognize the true meaning of the period doublings.

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