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In mathematics, the Gauss map (also known as Gaussian mapChaos and nonlinear dynamics: an introduction for scientists and engineers, by Robert C. Hilborn, 2nd Ed., Oxford, Univ. Press, New York, 2004. or mouse map), is a nonlinear iterated map of the reals into a real interval given by the
Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It i ...
: : x_ = \exp(-\alpha x^2_n)+\beta, \, where ''α'' and ''β'' are real parameters. Named after
Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...
, the function maps the bell shaped Gaussian function similar to the
logistic map The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popula ...
.


Properties

In the parameter real space x_n can be chaotic. The map is also called the ''mouse map'' because its
bifurcation diagram In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the sy ...
resembles a
mouse A mouse ( : mice) is a small rodent. Characteristically, mice are known to have a pointed snout, small rounded ears, a body-length scaly tail, and a high breeding rate. The best known mouse species is the common house mouse (''Mus musculus' ...
(see Figures).


References

Chaotic maps {{chaos-stub