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^[1] or mouse map), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function:

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

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.

References

1. Chaos and nonlinear dynamics: an introduction for scientists and engineers, by Robert C. Hilborn, 2nd Ed., Oxford, Univ. Press, New York, 2004.