# Advanced Topics - Demo 1

## Period Doubling in the Henon Map: small *b*

For small *b* the Henon map reproduces the dynamics of the quadratic map. The map equations can be rewritten

*
*
x_{n+1} |
= |
y_{n} + 1 - ax_{n}^{2} + bx_{n-1} |

y_{n} |
= |
bx_{n-1} |

so that for small *b* the first equation approximates the *one dimensional* quadratic map, and the second equation shows that *y*_{n} remembers the value of *x*_{n-1} (with the scale factor *b*) so that the Henon map will look like the plot of the quadratic map tipped on its side (or more precisely, the map function for regions of the interval that are attracting).
This is shown for *a=1.98* (which is equivalent to the value of 3.99 for the quadratic map parameter, i.e. the value giving chaos over the full unit interval) and *b=0.01*:

It should be noted that the curve is not strictly one dimensional - this can be seen better by enlarging the portion of the curve near the maximum - as must be the case for the invertibility of the Henon map to be preserved.

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Last modified Friday, February 4, 2000

Michael Cross