Can a LFSR has more than one repeat cycle? It's clear that an LFSR has finite states and eventually will end up with repeating cycle. However, is it possible for an LFSR to have more than one repeating cycle? For example, with initial state $A$, its goes to cycle $C_a$, and for initial state $B$, it ends up with cycle $C_b$.
There is a very nice recent paper available here by Chang, Ezermann, Ling and Wang, which has addressed this problem for arbitrary alphabet $\mathbb{F}_q$ and arbitrary characteristic polynomials, including those with divisors that are powers of irreducible polynomials.

Let $\Omega(f)$ be the set of sequences satisfying $f$ as their characteristic polynomial.

This paper determines the cycle structure of $\Omega(f)$ fully for an arbitrary $f$ and gives a method to identify a state from each distinct cycle in $\Omega(f)$, a longstanding open problem.