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:

• [math]\displaystyle{ E_c : z \to e^z + c }[/math]
• [math]\displaystyle{ E_\lambda : z \to \lambda * e^z }[/math]

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