The "Butterfly Effect" is often ascribed to the American meteorologist, Edward Lorenz, who developed in 1963 the system of 3 differential equations below:
For further details on the meaning of each equation please check Weisstein (refered below).
In this article Lorenz wrote 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 1972, in his speech delivered at the American Association for the Advancement of Science in Washington, D.C the seagull "became" a butterfly an left a question:
"Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?"
Initialization
>
restart;
>
with( DEtools ):
>
with( plots ):
>
with( linalg ):
>
with( PDEtools ):
Warning, the protected names norm and trace have been redefined and unprotected
Warning, the name changecoords has been redefined
Warning, the name adjoint has been redefined
Warning, the protected names norm and trace have been redefined and unprotected
Model Dynamics
>
eqs:={diff(X(t),t)=-sigma*X(t)+sigma*Y(t),
diff(Y(t),t)=-X(t)*Z(t)+r*X(t)-Y(t), diff(Z(t),t)=X(t)*Y(t)-b*Z(t)}:
>
iniset:=[[X(0)=-4,Y(0)=-2.5,Z(0)=15]]:
>
DEplot3d(eqs,{X(t),Y(t),Z(t)},t=0..180,X=-8..8,Y=-11..11,Z=4..24,iniset,stepsize=0.1, method=rkf45,linecolor=navy,thickness=1,axes=FRAME);
Powered by: JavaView
obs: If the graph doesn't come out press "reload" in your web browser. You must have enabled/instaled the
Java Virtual Machine.
To rotate, rescale or pick a vertex click the right-button and move your mouse.
The picture may take a few seconds to come out.
References:
- Lorenz, E. N. "Deterministic Nonperiodic Flow." J. Atmos. Sci. 20, 130-141, 1963.
- Weisstein, E. W. "Lorenz Attractor." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LorenzAttractor.html
