The readout would start off the same and then all of a sudden would no longer resemble the intended sequence.Īnimation of Baker transformation courtesy of Eviator Bach Though something bizarre would happen: He wouldn’t see the same sequence again. So that he could see sequences of results again, he would write down the row of numbers that occurred before the interesting part, and then the next time, he would start it off with those initial conditions from mid way through. The resulting sequence was printed out line by line as a row of twelve numbers. Edward Lorenz was heavily involved in that area and was equipped with a Royal McBee LGP-30 which he used to simulate weather patterns via a complicated set of ODEs. One field that benefited was numerical weather prediction. In the 1950s and 60s, the commercialisation of computers allowed for larger and heavier computations. SDIC was also the first part of chaos theory to be discovered. This is probably the most intuitive of the three conditions and makes the most sense when trying to relate to chaos. Sensitive dependence on initial conditions (SDIC) This has been an abusively short introduction to dynamical systems with (hopefully) just the bare-bones to understand this story. f applied to x₀ n times) and it is a periodic orbit if for some n, fⁿ(x₀) = x₀. The flow of x₀ at iterate n is simply fⁿ(x₀) (i.e. When we talk about a system, in this context we mean a dynamical system which (for our purpose) comes in two flavours: continuous time systems and discrete time systems.Ĭontinuous time systems are defined via an ordinary differential equation (ode)
GENTLE READER AMINITION FREE
If you have already met dynamical systems, feel free to skip this section and start reading again at Sensitive Dependence on Initial Conditions. But first I present: A crash course in dynamical systems In this article we will explore all three of these requirements and get to the root of what chaos truly means.
In Gleick’s book Chaos, the author interviewed a whole bunch of chaos scientists (chaosists?) and not one of them could agree on a definition of chaos.
Not because of the choice of axioms or some of the results, but due to the fact that no-one can agree what chaos actually means. Why the Butterfly Effect is not the whole storyĬhaos theory is potentially one of the most controversial fields in mathematics.
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