Exponential map (discrete dynamical systems) explained

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:

E _{c} : z \to e ^{z} + c E _{λ} : z \to λ * e ^{z}

The second one can be mapped to the first using the fact that

λ * e ^{z.} = e ^{z+ln(λ)} , so E _{λ} : z \to e ^{z} + 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. https://macau.uni-kiel.de/receive/diss_mods_00000781?lang=en Dynamics of exponential maps by Lasse Rempe
2. http://arxiv.org/abs/0805.1658 "Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity"