Lorenz Model

Model Description

This simple model was developed by Edward Lorenz in 1963 to study fluid mechanics (based on Navier-Stokes equations). It is the first ever model of a chaotic dynamical system. Chaos arises when a deterministic, nonlinear dynamical system exhibits long-term unpredictability in behavior due to sensitivity to initial conditions.

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.  When plotted in the three spatial dimensions, the dynamics of the model describe a shape called "strange attractor."

Default Dynamics

(Please describe default dynamics for this model.)

Lorenz

(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. 
#parameters
par s=10,r=27,b=2.666666
par c=.2, del=.1

#Initial conditions
init x=-7.5,y=-3.6,z=30

x'=s*(-x+y) 
y'=r*x-y-x*z 
z'=-b*z+x*y 

# x is proportional to the angular velocity, so integral is angle
theta'=c*x
th[0..7]=theta+2*pi*[j]/8
# approximate the velocity vector in the butterfly coords
z1=z-del*(-b*z+x*y)
x1=x-del*(s*(-x+y))

aux x1=x1
aux z1=z1

@ dt=.01, total=40, xplot=x,yplot=y,zplot=z,axes=3d
@ xmin=-20,xmax=20,ymin=-30,ymax=30,zmin=0,zmax=50
@ xlo=-1.5,ylo=-2,xhi=1.5,yhi=2,bound=10000
@ maxstor=20000
@ phi=60

done
# shows the waterwheel using the integrated ang velocity
# see Strogatz book  use lorenz2.ode
PERMANENT 
circ .515;.515;.46;$BLACK;2
TRANSIENT
SPEED 20
frect .5+.45*sin(th[0..7]);.5+.45*cos(th[j]);.55+.45*sin(th[j]);.55+.45*cos(th[j]);$BLACK
# plotting the butterfly and a lagged version of it !!
fcirc .5+x/40;z/50;.02;$GREEN
fcirc .5+x1/40;z1/50;.02;$RED
end

Yu JHD,  Arizona State University. 

Bozicevic M,  Arizona State University. 

Lorenz EN.  1963.  Deterministic nonperiodic flow. Journal of the Atmospheric Sciences. 20:130-141.

Lorenz
You can adjust total time of simulation below.
This model has four state variables. You can set their initial values below.
This model has five parameters. You can set their values below.
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