With "just" a linked list, you need a turtle (advance one-node at a time), and a hare (advance two-nodes at a time).
If the turtle and hare ever meet, you have a cycle. Otherwise, if the hare reaches the end of the list, you don't have a cycle.
If the turtle and hare ever meet, you have a cycle. Otherwise, if the hare reaches the end of the list, you don't have a cycle.
Another notable advantage of Brent's algorithm is that it automatically finds the cycle length, rather than (in Floyd's case) any multiple of the cycle length.
https://en.wikipedia.org/wiki/Cycle_detection#Brent's_algori...
Cycle detection in (currently) 44 languages. Most (all?) use Brent's algorithm. They're operating on an iterated function but converting most of these to detect cycles in a linked list would be straightforward.