Nonlinear Systems- 10 The Butterfly Effect

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In chaos theory, the butterfly effect is the phenomenon of sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. In this module we give an overview to the concept and its origins in meteorology with the famous Edward Lorenz computer experiment.

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Transcription excerpt:

Science is dependent upon isolating a system so as to study it, but this is literally impossible, everything has an effect on everything else in our universe, every single particle of matter has some gravitation effect on every other particle of matter, irrespective of how much we try to isolate something the concept of an isolated system that has zero interaction with its environment is really just a theoretical one that does not exist in reality. This simple insight literally destroys the entire scientific enterprise. Science is dependent upon this capacity to isolate systems, because if we want to say A causes B then we need to be able to control for all other variables, that is remove them from the equation. As we have stated this is not possible. So how do we get round this stumbling block, as it appears the scientific endeavor goes on without this causing too much of a problem. What we do, because we can’t fully isolate any system, is essentially define what are significant effects and what are negligible effect and simple forget about the negligible effects. So for example if I am doing some research in my lab on the interaction of two particles of matter, under this premise I do not need to take into account the gravitation effect that some planet the other side of the universe is having on these particles because it is deemed negligible. This basic premise that small causes can only cause relatively small effects is one of the basic assumptions and principals that gives our world some order, we depend upon it almost all day every day, I feel confident in the fact that if I forget to pay my bank overdraft this week it is not going to bring the whole global economy into melt-down, or that a teeny little pin prick is not going to kill me. We find order in the world in the fact that the chances of these phenomena happening are so small, that they are negligible and we can thus forget about them, without this being the case within linear systems our world would be extraordinary random and chaotic. But as we have previously discussed, nonlinear systems through feedback loops, can grow exponentially, this means that negligible effects or differences, within nonlinear systems can themselves grow in an exponential fashion where small effects and errors are fed back into the system at each stage of its development to compound the size of the errors as it grows at an exponential fashion. As was the case in the famous Edward Lorenz computer experiment, where when he fed values into the computer that he thought were exactly the same, the output results the computer give him were widely divergent. He eventually traced this back to small differences in rounding errors, that made the values only very slightly different, but through iteration these very small errors would grow not in a linear fashion but exponentially making the resulting output widely divergent within a relatively short period of time and thus giving us the phenomena that is called sensitivity to initial conditions.

Sensitivity to initial conditions is popularly known as the "butterfly effect", thought to be so called because of the title of a talk given by Edward Lorenz in 1972 called Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to some large-scale phenomena. Had the butterfly not flapped its wings, the future trajectory of the system might have been vastly different.

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Complexity Academy

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