# The Butterfly Effect

The "Butterfly Effect", or more technically the "sensitive dependence on initial conditions", is the essence of chaos. This is illustrated in the accompanying applet of the Lorenz Attractor.

The demonstration shows a graphical representation of the time variation of three variables X(t),Y(t) and Z(t), coupled by non-linear evolution equations. For the default parameters of the applet a single solution is shown evolving from an initial condition (X0,Y0,Z0). You can also start two solutions running simultaneously from initial conditions separated by (dX0,dY0,dZ0) by setting any of dX0, dY0, dZ0 to nonzero values (e.g. 0.01). This tiny difference in the initial conditions becomes amplified by the evolution, until the two trajectories evolve quite separately. The amplification is exponential, the difference grows very rapidly and after a surprisingly short time the two solutions behave quite differently. This is an illustration of the butterfly effect - the idea in meteorology that the flapping of a butterfly's wing will create a disturbance that in the chaotic motion of the atmosphere will become amplified eventually to change the large scale atmospheric motion, so that the long term behavior becomes impossible to forecast.

The "Butterfly Effect" is often ascribed to Lorenz. In a paper in 1963 given to the New York Academy of Sciences he remarks:

One meteorologist remarked that if the theory were correct, one flap of a seagull's wings would be enough to alter the course of the weather forever.
By the time of his talk at the December 1972 meeting of the American Association for the Advancement of Science in Washington, D.C. the sea gull had evolved into the more poetic butterfly - the title of his talk was* :
Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?
In the applet we also see a second incarnation of the Butterfly - the amazing geometric structure discovered by Lorenz in his numerical simulations of three very simple equations that now bear his name.
* As quoted in "Chaos and Nonlinear Dynamics" by R.C.Hilborn (Oxford Uni-versity Press, 1994). This information was kindly sent to me by Corrie Modell.