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=ac 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 2n 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 2n cycle.
- The Lyapunov exponent first becomes positive at a=ac: this can be identified as the onset of chaos.
- Even for a>ac there are regions of negative Lyapunov exponent corresponding to the periodic bands.
Last modified Tuesday, December 7, 1999