Gauss iterated map

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_{n}^{2})+\beta ,\,

Cobweb plot of the Gauss map for

\alpha =4.90

and

\beta =-0.58

. This shows an 8-cycle.

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

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

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