In mathematics, the Gauss map (also known as Gaussian map^[1] or mouse map), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function:

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

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.

In the parameter real space ${\displaystyle x_{n}}$ can be chaotic. The map is also called the mouse map because its bifurcation diagram resembles a mouse (see Figures).