# The Lorenz Model - Demo 7

## Return Map

Lorenz noted that the maximum values of the *Z* variable obtained on successive orbits around
one or other fixed point seemed to have some predictability. He therefore
plotted a return map of successive maximum values of this variable i.e. plot the *(n+1)*th value of *Z*_{max} against the *n*th value of *Z*_{max}

The return map (that is filled in after some long iteration time) has the appearance of a
function (i.e. is single valued), and leads us later to study "one dimensional maps". (Again,
looking on a very fine scale we would find that the points do not strictly define a function -
there is some slight "fuzziness" to the line.) A great deal of information about the Lorenz
dynamics can be intuitively understood from the return map. For example the intersection of
the "function" with the diagonal line corresponds to a fixed point; the fact that the magnitude
of the slope of the function is greater than unity shows that the fixed point is unstable. Indeed
we can associate the chaotic motion with the fact that the magnitude of the slope is greater
than unity everywhere. This leads us to the idea of the "sensitivity to initial consitions".

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Last modified Thursday, November 18, 1999

Michael Cross