# 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]}

## 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:

The second one can be mapped to the first using the fact that , so is the same under the transformation . 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

- ↑ Dynamics of exponential maps by Lasse Rempe
- ↑ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity

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