# One Dimensional Maps - Demo 5

## Lyapunov Exponent for the Quadratic Map

The Lyapunov exponent for the quadratic map is shown as a function of *a* here. After viewing the overall curve, you might wish to enlarge the interesting region near *a=a*_{c} by dragging the mouse over the region of interest on the stationary plot. To investigate the fine details you will need to increase *Points* and *Transient*.

Some evident features are:

- Values of
*a* for which the Lyapunov exponent is zero: these are the bifurcation points where an orbit just becomes unstable
- Values of
*a* for which the Lyapunov exponent diverges to negative infinity: these are where the period 2^{n} orbit exactly hits the maximum of the map, where the slope is zero. These are known as "superstable" cycles, and provide a conveniently identified point within the stability range of each 2^{n} cycle.
- The Lyapunov exponent first becomes positive at
*a=a*_{c}: this can be identified as the onset of chaos.
- Even for
*a>a*_{c} there are regions of negative Lyapunov exponent corresponding to the periodic bands.

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Last modified Tuesday, December 7, 1999

Michael Cross