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 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=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.