Model Description
Animated demonstration of the Lorenz model and its sensitivity to initial conditions. The simulation starts with twenty points very close to each other, and follows them as they move further away. The starting values differ in the fifth and sixth significant digits of a single coordinate.
The Lorenz model was developed by Edward Lorenz in 1963 to study fluid mechanics.The model is a three-dimentional system of differential equations. Specifically, the model describes the convection motion of a fluid in a small idealized "Rayleigh-Benard" container. The basic presumption is that there is fluid in a container with surfaces at different temperatures. The model shows that the fluid motion eventually leads to a chaotic behavior.
Reference
Lecture notes by David Marshall (Professor at University of Oxford). Link: http://www.atm.ox.ac.uk/user/marshall/ffc/lecture13.pdf
Default Dynamics
(Please describe default dynamics for this model.)
Lorenz 20
(Please describe this animation.)
$\frac{dx}{dt} = \sigma (y - x)$ |
| Rate of change in the strength of the convective motion. |
$\frac{dy}{dt} = x (\rho - z) - y$ |
| Rate of change in the temperature difference between the ascending and descending currents. |
$\frac{dz}{dt} = xy - \beta z$ |
| Rate of change in the distortion of the vertical temperature profile from linearity. |
# 20 lorenz equations x[21..40]'=s*(y[j]-x[j]) y[21..40]'=r*x[j]-y[j]-x[j]*z[j] z[21..40]'=-b*z[j]+x[j]*y[j] par r=27,s=10,b=2.66 init x[21..40]=-7.5,y[j]=-3.6 init z[21..40]=30.00[j] @ total=50,dt=.02 done
# animation fcirc (x[21..40]+20)/40;z[j]/50;.02;2-[j]/20 done
Bozicevic M, Arizona State University.
. 1963. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences. 20:130-141.