Cobweb plot of the Gauss map for
α α -->
=
4.90
{\displaystyle \alpha =4.90}
β β -->
=
− − -->
0.58
{\displaystyle \beta =-0.58}
In mathematics , the Gauss map (also known as Gaussian map [ 1] mouse map ), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function :
x
n
+
1
=
exp
-->
(
− − -->
α α -->
x
n
2
)
+
β 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}}
mouse map because its bifurcation diagram resembles a mouse (see Figures).
Bifurcation diagram of the Gauss map with
α α -->
=
4.90
{\displaystyle \alpha =4.90}
β β -->
{\displaystyle \beta }
Bifurcation diagram of the Gauss map with
α α -->
=
6.20
{\displaystyle \alpha =6.20}
β β -->
{\displaystyle \beta }
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 Chaoticlist )
Physical Chaos Related
