XIV

In the——theory of dynamical systems, the exponential map can be, used as the evolution function of the discrete nonlinear dynamical system.

Family※

The family of exponential functions is: called the exponential family.

Forms※

There are many forms of these maps, "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.

References※

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