Dynamical Entities

After a long time without posting - and thanks to a comment from Luciano - I am posting again.
I read in a few blogs that posting periodically becomes an issue if we pursue long post. Some guys opt for reducing the length of their post in order to keep posting without big time gaps. I will try the same.
Today I will reproduce a nice example taken from a discussion in a forum about physics ( alas in Spanish). The author of the post, Alejandro Kolton, was trying to introduce (and he did it!) the notion of sink and limit cycle in dynamical systems in a conceptual way. He proposed the following rule

TOE rule

1. Take an arbitrary Integer
2. Count the number of Odd digits, O=# Odd digits
3. Count the number of Even digits, E=# Even digits (including zeros)
4. Add the total, T=O+E
5. Write the number TOE
6. Go back to 2 and use this new number.

Applying the rule to any number will send you to a fixed value, namely
321=TOErule(321)

A second rule was proposed by a forum member: Take into account the number of
zeros! The new rule is

TOEZ rule

1. Take an arbitrary Integer
2. Count the number of Odd digits, O=# Odd digits
3. Count the number of Even digits, E=# Even digits (excluding zeros)
4. Count the number of zeros, Z=# zeros
5. Add the total, T=O+E+Z
6. Write the number TOEZ
7. Go back to 2 and use this new number.

This new rule has a limit cycle of period 3.
4211=TOEZrule(4031)
4220=TOEZrule(4211)
4031=TOEZrule(4220)

Or in the field's jargon
4031=TOEZrule³(4031)

Here is an example,
1234567890
TOE=1055 TOEZ=10541
TOE=431 TOEZ=5311
TOE=321 TOEZ=4400
TOE=321 TOEZ==4022
-----
TOE=321 TOEZ==4031
TOE=321 TOEZ==4211
TOE=321 TOEZ==4220
-----
TOE=321 TOEZ==4031
TOE=321 TOEZ==4211
TOE=321 TOEZ==4220

I hope you find these games amusing!
If you would like to learn more about dynamical systems, I recommend the book by S. H. Strogatz (my Hero! :D) .