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Butterfly Effect

The Lorenz attractor shows how some small change in initial conditions can lead to a large change in the system's long-term trajectory

The Lorenz attractor shows how some small change in initial conditions can lead to a large change in the system’s long-term trajectory

The butterfly effect is a popular term for what is called sensitive dependence on initial conditions in which a small change in some input state to a nonlinear system can result in a vastly disproportionate output at a later stage.1 The term is thought to derive from 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 butterfly effect is part of the broader area of chaos theory that studies the dynamics of nonlinear systems (in particular those that are sensitive to initial conditions). After having gained acceptance within mainstream mathematics and science during the seventies and eighties, the butterfly effect has been since identified in many different areas from weather patterns, to supply chains networks, to electrical power grids and international politics.

Cause & Effect

Everything has an effect on everything else in our universe. Every single particle of matter has some gravitational 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. Our traditional conception of 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 all other variables, that is remove them from the equation. As we have stated this is not possible. So how do we get around 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 not fully isolate any system, is essentially define what are significant effects and what are negligible effects and simply forget about the negligible effects. 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 gravitational effect that some planet on 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 principles that gives our world some order. We depend on 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 meltdown, 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 extraordinarily random and chaotic.

Feedback Loops

Nonlinear phenomena like hurricanes are driven by feedback loops

Nonlinear phenomena like hurricanes are driven by feedback loops

In nonlinear systems, though, 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 in 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 gave 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 phenomenon 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. Something to note is that the butterfly does not directly cause the tornado. This is, of course, impossible. The flap of the butterfly’s wings simply defines some initial condition. It is then the set of chain reactions through feedback loops that enable a small change in the initial conditions of the system to have a significant effect on its output, rendering long-term predictions virtually impossible.

Nonlinearity

With respect to the unpredictable nature of the butterfly effect, one might say, well if our initial measurement is wrong then obviously our prediction of its future state is also going to be wrong! But this is missing the point, which is firstly that this inaccuracy is growing exponentially as the system develops – it is not just staying the same, and secondly that in these nonlinear systems we can never know exactly the starting condition as our accuracy of measurement must grow in an exponential fashion in order to achieve just a linear growth in our horizon of predictability. Chaos and the butterfly effect after being shunned by the scientific community for many decades are today accepted as scientific facts, a fundamental and inescapable part of the dynamics of nonlinear systems. They show again how when things can grow exponentially we can get extraordinary and counter-intuitive results.

Cite this article as: Joss Colchester, "Butterfly Effect," in Complexity Academy, May 5, 2015, http://complexityacademy.io/butterfly-effect/.
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