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Cobweb plot of the Gauss map for
α
=
4.90
{\displaystyle \alpha =4.90}
and
β
=
−
0.58
{\displaystyle \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
(
−
α
x
n
2
)
+
β
,
{\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 .
Properties
In the parameter real space
x
n
{\displaystyle 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
α
=
4.90
{\displaystyle \alpha =4.90}
and
β
{\displaystyle \beta }
in the range −1 to +1. This graph resembles a mouse.
Bifurcation diagram of the Gauss map with
α
=
6.20
{\displaystyle \alpha =6.20}
and
β
{\displaystyle \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.
Concepts
Theoretical branches Chaotic maps (
list )
Physical systems Chaos theorists Related articles