Lorenz Notes

Lorenz Mask Notes

Lorenz Mask, projected onto XZ plane
from www.zeuscat.com/andrew/chaos/lorenz.html

-- the Lorenz Mask represents the solution to a three-dimensional set of ordinary differential equations which can model the unpredictable behavior of the weather; found by E. Lorenz of MIT;

dx/dt = P(y-x)
dy/dt = Rx-y-xz
dz/dt = xy-Kz

where x is the rate of convective overturning,
y is the horizontal temperature overturning,
z is the vertical temperature overturning;

and P is proportional to the Prandtl number (ratio of the fluid viscosity of a substance to its thermal conductivity, usually set at 10)
R is proportional to the Rayleigh number (difference in temperature between the top and bottom of the system, usually set at 28)
K is a number proportional to the physical proportions of the region under consideration (width to height ratio of the box which holds the system, usually set at 8/3)

-- all three numbers are positive since they all represent physical quantities
-- convection is the process by which heat is transferred by a moving fluid; Mask can be derived from observing the convective motion in a 2-D fluid cell that is warmed from below and cooled from above; though it seems regular, this motion is actually chaotic
-- Ludwig Prandtl is a German physicist who is considered to be the father of aerodynamics
-- John William Strutt Rayleigh is an English physical scientist who made discoveries in the fields of optics and acoustics that are essential to the theory of wave propagation in fluids

Observations:

trajectory does not intersect itself in three dimensions
trajectory is not periodic or transient
general form of the shape does not depend on initial conditions
exact sequence of loops is very sensitive to the initial conditions

Properties:

symmetry: (x, y, z) --> (-x, -y, z) for all values of the parameters
the z-axis (x=y=0) is invariant (all trajectories that start on it also end on it)

Interesting application:

-- Lorenz Mask used as an encryption method for masking information-bearing waveforms

By Yvan Gauthier of Carleton University, April 1998.

-- On the transmitting end, an information-bearing signal i(t) is added to an evolving Lorenz dynamical system x to produce the output s(t), which is the masked signal.
s(t) = x + i(t)
-- On the receiving end, the process is reversed, and x is subtracted from s(t), (hopefully) recovering the original signal I(t).
I(t) = s(t) - x

-- Though this process does seem to work, it is highly inefficient for periodic and high-frequency signals, but can be used to mask other signals (such as voice signals).

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