How to compare two circular sequences in linear time?
Start with computing the lexicographlcally least circular substrings$^1$ of the both circular sequences and then compare them directly.

Alternatively you can check for the substring $A$ (the first seqience) in the string $BB$ (a concatenation of $B$, the second sequence) circular sequence with itself) using for example the KMP$^2$

You might also be interested in the application of the Suffix Trees (also Suffix Arrays) and this thesis reviewing the applications to DNA sequences.

$^{[1]}$(described in K. S. Booth. Lexicographically least circular substrings.
Inf. Process. Lett., 10(4/5):240-242, 1980., and here
$^{[2]}$(Donald E. Knuth, James H. Morris, Jr., and Vaughan R. Pratt,  SIAM J. Comput., 6(2), 323–350. Fast Pattern Matching in Strings