Gauss iterated map

Cobweb plot of the Gauss map for $\alpha =4.90$ and $\beta =-0.58$. 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:

$x_{n+1}=\exp(-\alpha x_{n}^{2})+\beta ,\,$

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 $x_{n}$ 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 $\alpha =4.90$ and $\beta$ in the range −1 to +1. This graph resembles a mouse.

Bifurcation diagram of the Gauss map with $\alpha =6.20$ and $\beta$ in the range −1 to +1. This graph resembles a mouse with slightly bigger ears.