Gauss iterated map

Cobweb plot of the Gauss map for [math]\displaystyle{ \alpha=4.90 }[/math] and [math]\displaystyle{ \beta=-0.58 }[/math]. This shows an 8-cycle.

In mathematics, the Gauss map (also known as Gaussian map or mouse map), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function:

[math]\displaystyle{ x_{n+1} = \exp(-\alpha x^2_n)+\beta, \, }[/math]

where α and β are real parameters.

Named after Johann Carl Friedrich Gauss, the function maps the bell shaped Gaussian function similar to the logistic map.

Properties

In the parameter real space [math]\displaystyle{ x_n }[/math] can be chaotic. The map is also called the mouse map because its bifurcation diagram resembles a mouse (see Figures).

Bifurcation diagram of the Gauss map with [math]\displaystyle{ \alpha=4.90 }[/math] and [math]\displaystyle{ \beta }[/math] in the range −1 to +1. This graph resembles a mouse.

Bifurcation diagram of the Gauss map with [math]\displaystyle{ \alpha=6.20 }[/math] and [math]\displaystyle{ \beta }[/math] in the range −1 to +1.