How to produce nonzero absolute differences between neighboring numbers on a circle as long as possible?
I am a bit cautious just in case you might be taking part of some kind of open competition or exam.

Before I could reasonably believe that you are raising this question in good faith, I will just say I can reach 51 iterations without going to all zeros using less than 18 digits for each number.

As an evidence that I do have an answer, here are the last 5 digits of my numbers. This will not help anybody to get a better answer, except when you are super good at math, in which situation you probably do not need this too tenuous a hint anyway. However, it can be used to prove beyond any doubt in the future that I did know an answer.
$$(***00000, ***22634, ***22881, ***60047)$$



Per request of the questioner, I will explain the simple "secret" of how to get an answer briefly. A full update may or may be followed later.

What you want is to let $a_1, b_1,c_1,d_1$ look like $a_0, b_0,c_0,d_0$ exactly. Since they look like exactly, the $a_2, b_2, c_2, d_2$ will look like $a_1, b_1, c_1, d_1$ exactly. You see you can actually go on forever. It will break down only when we cannot express the number accurately enough in its numerical representation, which is enforced since the geocacher owner does not accept values like $\sqrt 2$ (just for an example; there may or may not be a $\sqrt 2$ in my answer).

What does it mean by exactly alike? Well, for example, if the two circles of numbers are the same except by a factor and a rotation.