This method cannot be applied to Collatz due to the factor 3 whenever an odd term is encountered. But it can be applied to another variant where odd n \rightarrow \frac{3n+1}{2} and even n \rightarrow \frac{3n}{2}. This is an equivalent formulation for the “tricot” 2n \rightarrow 3n and 2n+1 \rightarrow 3n+2, already discussed in the topic Finding a specific irrational number.
Obviously, all positive integers have divergent trajectories. But, the situation is less clear on (negative) rationals with an odd denominator. Starting from -\frac13 or -\frac23 leads to the fixed points 0 and -1, respectively. The previous method implies that the length of a rational cycle with denominator q \geq 5 and coprime to 3 is given by the order of \frac32 in \mathbb{Z}_q^{\rm x}. However, the existence of such a cycle is not guaranteed. For q=5, we have the cycle -\frac25 \rightarrow -\frac35 \rightarrow -\frac25 of period 2, but there seems to be no cycle for q=7 and q=11. There is also a cycle for the denominator 13 with period 4, starting from -\frac{4}{13}.
One may ask whether there are infinitely many cycles and what periods are possible (or not)?