In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Gauss map (also known as Gaussian map
[Chaos 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 (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...
:
:
where ''α'' and ''β'' are real parameters.
Named after
Johann Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
, the function maps the bell shaped Gaussian function similar to the
logistic map
The logistic map is a discrete dynamical system defined by the quadratic difference equation:
Equivalently it is a recurrence relation and a polynomial mapping of degree 2. It is often referred to as an archetypal example of how complex, ...
.
Properties
In the parameter real space
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 ...
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