With 2 beads RB changes to BB which changes to RR (the steady state) so this always reduces to a steady state.

With 3 beads the only initial states are RRR BBB RBB and BRR

(1) With 3 beads the steady state is RRR
(2) BBB becomes RRR and then remains RRR for all time (3) RBB becomes BRB then BBR then RBB again giving a periodic 3-cycle (4) BRR becomes BRB goes into a periodic 3-cycle as in (3)

Generalising this note that for any number of beads if all the beads are red RRR...R this is a steady state. If all the beads are blue then at the first step they all become red and reach the steady state.

To encode this process we can write 0 for the red beads and 1 for the blue beads and denote the sequence of n beads by the vector ( x0 ,  x1 , ... xn ) where all the xs are 0 or 1.
This vector is mapped to ( x0 + x1 , x1 + x2 ,... xn + x0 ) where addition is modulo 2. Note that 1 + 1 =0 and 0 + 0 = 0 which corresponds to replacing two of the same colour by a red.
Also 1 + 0 = 1 and 0 +1 = 1 which corresponds to replacing two of different colours by a blue.

For 4 beads there are 6 positions which all reduce to the steady state with all red beads.

(1) 1111 → 0000 (6)
(2) 0111 → 1001 (3) → (4) → (1) → (6)
(3) 0011 → 0101 (4) → (1) → (6)
(4) 0101 → 1111 (1) → (6)
(5) 0001 → 0011 (3) → (4) → (1) → (6)
(6) 0000 → 0000 (6)

For 5 beads there are 8 positions two reducing to the steady state with all red beads and the other six cycling between 3 positions.

(1) 11111 → 00000 (8)
(2) 01111 → 10001 (5)(6)(2) which is a 3-cycle
(3) 00111 → 01001 (6) then into the 3-cycle (2)(5)(6)
(4) 01011 → 11101 (2) then into the 3-cycle (5)(6)(2)
(5) 00011 → 00101 (6)(2)(5) again the 3-cycle
(6) 00101 → 01111 (2)(5)(6) again the 3-cycle
(7) 00001 → 00011 (5) then into the 3-cycle (6)(2)(5)
(8) 00000 → 00000 (8) which is the steady state

For 6 beads there are 13 positions three reducing to the steady state with all red beads, three reducing to a steady state with 2 red and 4 blue beads nd the other seven eventually cycling between 2 positions.

(1) 111111 → 000000 (13)
(2) 011111 → 100001 (9)(10)(3) the 2-cycle
(3) 001111 → 010001 (10)(3) is a 2-cycle
(4) 010111 → 111001 (3)(10) the 2-cycle
(5) 011011 → 101101 (5) which is a steady state
(6) 000111 → 001001 (11)(5)which is a steady state
(7) 001011 → 011101 (4)(3)(10)the 2-cycle
(8) 010101 → 111111 (1)(13) the red steady state
(9) 000011 → 000101 (10)(3) the 2-cycle
(10)000101 → 001111 (3)(10) the 2-cycle
(11)001001 → 011011 (5) which is a steady state
(12)000001 → 000011 (9)(10)(3) the 2-cycle
(13)000000 → 000000 (13) which is the steady state with all red beads