One Dimensional Maps - Demo 6

Other Maps

A number of other one dimensional maps may be reviewed in the applet. It is useful to study the "bifurcation map" first, and then to pursue particular phenomena in more detail. Appropriate maps to look at at this stage are:

• The sine map: a map with a quadratic maximum but different functional form to the quadratic map: shows the same range of phenomena as the quadratic map, and in fact there are "universal" quantitative features of the "period doubling route to chaos" that apply to both maps (and other maps with a quadratic maximum).

• The tent map: this piecewise linear map allows many quantities to be calculated analytically. For example the Lyapunov exponent is log a and becomes positive at a=1. The tent map at a=2 is equivalent to the Bernoulli shift map.

• The "power law" map interpolates between the tent map and the quadratic map: it has a "maximum" at x=0.5 with a power law cusp here determined by the parameter b: b=0 yields the tent map and b=1 the quadratic map. Perturbing in small b is a useful technique for understanding the period doubling route to chaos. Here the behavior is shown for b=0.5

(The circle map will be discussed later).

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