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.¹

The family of exponential functions is called the exponential family.

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.