We study the periodic solutions of the non--autonomous periodic
Lyness' recurrence u_{n+2}=(a_n+u_{n+1})/{u_n}, where a_n is a cycle with positive values a,b and with positive initial conditions. Among
other methodological issues we give an outline of the proof of the
following results: (1) If (a,b)\neq(1,1), then there exists a value
p_0(a,b) such that for any p>p_0(a,b) there exist continua of
initial conditions giving rise to 2p--periodic sequences. (2) The
set of minimal periods arising when a and b are positive and
positive initial conditions are considered, contains all the even
numbers except 4, 6, 8, 12 and 20. If a\neq b, then it does not
appear any odd period, except 1.
We study the periodic solutions of the non–autonomous periodic Lyness’ recurrence un+2 = (an +un+1)=un, where fangn is a cycle with positive values a,b and with positive initial conditions. Among other methodological issues we give an outline of the proof of the following results: (1) If (a;b) 6= (1;1), then there exists a value p0(a;b) such that for any p > p0(a;b) there exist continua of initial conditions giving rise to 2p–periodic sequences. (2) The set of minimal periods arising when (a;b) 2 (0;¥) 2 and positive initial conditions are considered, contains all the even numbers except 4, 6, 8, 12 and 20. If a 6= b, then it does not appear any odd period, except 1.