Exponential map (discrete dynamical systems)

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In the theory of dynamical systems, the exponential map can be used as the evolution function of the discrete nonlinear dynamical system.[https://macau.uni-kiel.de/receive/diss_mods_00000781?lang=en Dynamics of exponential maps by Lasse Rempe]

Family

The family of exponential functions is called the exponential family.

Forms

There are many forms of these maps,[http://arxiv.org/abs/0805.1658 "Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity"], Lasse Rempe, Dierk Schleicher many of which are equivalent under a coordinate transformation. For example two of the most common ones are:

$E_c : z \to e^z + c$
$E_\lambda : z \to \lambda * e^z$

The second one can be mapped to the first using the fact that $\lambda * e^z. = e^{z+ln(\lambda)}$ , so $E_\lambda : z \to e^z + ln(\lambda)$ is the same under the transformation $z=z+ln(\lambda)$ . The only difference is that, due to multi-valued properties of exponentiation, there may be a few select cases that can only be found in one version. Similar arguments can be made for many other formulas.

References

Category:Chaotic maps