Fractals/exponential

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

parameter space of the complex exponential family f(z)=exp(z)+c. The parameter in the middle of the picture is postsingularly preperiodic (PSP). Eight parameter rays landing at this parameter are drawn in black. The bifurcation locus is grey, while hyperbolic components are shown as colored regions.

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:

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

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

How to compute it

$Z=x+y*i$

$\exp(Z)=e^{Z}$

$\mathrm {Real(\exp(Z))} =\exp(x)\cos(y)$ 

$\mathrm {Imag(\exp(Z))} =\exp(x)\sin(y)$

What is the continous iteration of $e^{x}-1$ ?

"The function

 $e^{x}-1$ 

is one of the simpler applications of continuous iteration. The reason why is because regular iteration requires a fixed point in order to work, and this function has a very simple fixed point, namely zero: "

 $e^{0}-1=0$

Images

• commons Category:Exponential maps

See also

• Julia_and_Mandelbrot_sets_for_transcendental_functions by Gertbuschmann
• Exponential mapping of the plane
• Exponential maps

• tetration fractals
• Baker domains