Exponential map (discrete dynamical systems)

Parameter plane of the complex exponential family f(z)=exp(z)+c with 8 external ( parameter) rays

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:

* Ec:z->ez+c
* Eλ:z->λ*ez

The second one can be mapped to the first using the fact that λ*ez.=ez+ln(λ), so Eλ:z->ez+ln(λ) is the same under the transformation z=z+ln(λ). 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

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

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* Arnold tongue
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* Complex quadratic map
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* Duffing equation
* Duffing map
* Dyadic transformation
* Dynamical billiards
* outer
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* Exponential map
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* Gingerbreadman map
* Hénon map
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* Ikeda map
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* Kaplan–Yorke map
* Logistic map
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* Bouncing ball dynamics
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