And while the applet is loading..........

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:

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 wasOne 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.

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.Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?

* 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.

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Last modified August 18, 2009

Michael Cross