# Unkown Blogger Pursues a Deranged Quest for Normalcy

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Posted by ubpdqn on July 18, 2012

This post is motivated/inspired by Professor Bender’s book and lectures.  The following graphics relate to the continued exponential. The color coding is a first attempt at color coding the orbits of the iterated exponential:

\$latex a_0=1\$

\$latex a_n=e^{z a_{n-1}}\$

For example, \$latex a_3=e^{ze^{ze^z}}\$

The cardioid delimits the zone of convergence (to unique limit). It’s mottled appearance  is due to areas of convergence slower than my arbitrarily chosen iteration number (and they way I scripted partitioning the  orbits). The other colors are orbits of varying oscillatory/cyclic behaviour (varying points). The color schemes are the Gradient schemes (51) from Mathematica.  The distribution of limit cycle size was skewed so the cycle  size was logarithmically transformed to display the  range.

The animated  gif is lower resolution grid of complex plane [-2,2] x [-2,2]. The higher resolution images take longer to generate and with time I hope to write a better algorithm, chose a…

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