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In simple terms the system has gone out of control. There is no way to predict its long time behaviour. The surprise: The sequence is determined by its initial value - and yet, it cannot be predicted other than by letting it run. The problem is that any real description of the initial size of the sequence, its representation in a computer for instance, can only be given with finite precision. The process can be viewed as an unfolding of information: the longer we observe it, the better we know.
The most exciting aspect is not the chaos as such, but the scenario by which the order turns into chaos.
An exact analysis of the bifurcation points (the exact growth parameters for the oscillation between two periods) in the scenario shows that the doubling factor approaches a universal value of fÔ = 4.669201 ... as the period increases. This number is called the ¡§Feigenbaum number¡¨, because Mitchell Feigenbaum discovered the universality of this number.
The discovery has spurred an enormous activity among scientist of many fields. Mathematicians for example are still trying to fully understand that unexpected universality. But perhaps more important it has boosted a general hope that non - linear phenomena may not be out of reach of systematic scientific classifications.
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