# Fractals/exponential

In the theory of dynamical systems, the exponential map can be used as the evolution function of the discrete nonlinear dynamical system.[1]

## Family

The family of exponential functions is called the exponential family.

## Forms

There are many forms of these maps,[2] many of which are equivalent under a coordinate transformation. For example two of the most common ones are:

• ${\displaystyle E_{c}:z\to e^{z}+c}$
• ${\displaystyle E_{\lambda }:z\to \lambda *e^{z}}$

The second one can be mapped to the first using the fact that ${\displaystyle \lambda *e^{z}.=e^{z+ln(\lambda )}}$, so ${\displaystyle E_{\lambda }:z\to e^{z}+ln(\lambda )}$ is the same under the transformation ${\displaystyle z=z+ln(\lambda )}$. 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.

# How to compute it

${\displaystyle Z=x+y*i}$

${\displaystyle \exp(Z)=e^{Z}}$

${\displaystyle \mathrm {Real(\exp(Z))} =\exp(x)\cos(y)}$

${\displaystyle \mathrm {Imag(\exp(Z))} =\exp(x)\sin(y)}$


# What is the continous iteration of ${\displaystyle e^{x}-1}$ ?

"The function

 ${\displaystyle e^{x}-1}$


is one of the simpler applications of continuous iteration. The reason why is because regular iteration requires a fixed point in order to work, and this function has a very simple fixed point, namely zero: "[3]

 ${\displaystyle e^{0}-1=0}$