Exponential map (discrete dynamical systems)
# Exponential map (discrete dynamical systems)

In the theory of dynamical systems, the exponential map can be used as the evolution function of the discrete nonlinear dynamical system.

## Family

The family of exponential functions is called the exponential family.

## Forms

There are many forms of these maps, 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.

