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:

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

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.

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