Analysis of piecewise-defined sequences with an Integer domain

Why?

The analysis of piecewise-defined sequences is essential to the investigation of properties of numbers, and it holds interesting artistic value for sequences that alternate in a closed loop.
There are multiple interesting properties of numbers, such as that a number is divisible by 3 if the sum of the digits is divisible by 3, or if the last digit is even, then the whole number is divisible by 2.
In the same way, I think that investigation of such sequences would not only solve (or to be precise, explain) the Collatz Conjecture, but it is a cool problem to work on computationally for even first-year undergrads and high school students, and PhD students alike. It is interesting in the sense that you can play with the problem and get interesting results.

The plan

Let us divide the Integers in any way you wish. For example, we can define two subsets such that one subset would contain all the composite numbers (C), and the other one would contain the all the prime numbers (P), and so P union C equals the integers.
In a similar fashion, we can define three subsets such that S1 would have all numbers divisible by 3 (ie. congurent zero mod 3), S2 would be all integers congurent 1 mod 3, and S3 would be all integers congurent 2 mod 3, and S1 union S2 union S3 is all of the integers.

Then, let us define an equation (polynomials, probably) for each subset to form the piecewise function. For example, we define S1 to be all numbers divisible by 2, and S2 to be all numbers congurent 1 mod 2, then for S1 we define a function 3x+1, and for S2 we define a function .5x (Collatz, wink wink).
We have two main questions to answer here: What are the starting values of such sequences that would circle back eventually to the same value, and what are the values of such sequences that would diverge to infinity (or zero/one).

The challenge here, is to find these values preferably without brute forcing. But most importantly, to find why they behaved in that way, and what are the common properties between these values.
Also, how these values relate to how we split the integers and how we chose the equations to be related to each partition of the integers.

So, what do I need to do?

Construct some sequences, play with them. If you found anything interesting, please send me an email and I will publish it here. I will typically create a page for each sequence generalization and provide a timeline for all findings related to that generalization.