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

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

• Chaotic maps
• Mathematical analysis stubs
• Chaos theory stubs

• Commons category link is on Wikidata
• All stub articles