On Periodic Solutions of 2-Periodic Lyness' Equations

Abstract

We study the existence of periodic solutions of the nonautonomous periodic Lyness' recurrence u n+2 = (a n + u n+1 )/u n , where {a n } n is a cycle with positive values a, b and with positive initial conditions. It is known that for a = b = 1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a, b) = (1, 1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a = b, then any odd period, except 1, appears.

On the Set of Periods of the 2-Periodic Lyness’ Equation

“…As mentioned before (see also [1,Lemma 22]) it is proved that if a = b there are no period orbits of odd period, and that (a, b) = (1, 1) is the only point in the parameter space satisfying both that σ (a, b) = 2/5 and…”

“…It is also known (see [6], and also [1]) that the dynamics of F b,a restricted on each connected component of the invariant curves C h is conjugate to a rotation on the unit circle.…”

“…We will only outline the main results and focus on some methodological issues. We refer the reader to [1], where a further analysis and the proofs of the main results can be found.…”

“…First observe that since F b,a is conjugated to a rotation on each set C In [6] (and also in [1] using another approach) is proved that θ b,a (h) is a continuous function in [h c , ∞), in fact analytic in (h c , ∞). Furthermore, we can compute θ b,a (h c ) := lim h→h c θ b,a (h) and lim h→+∞ θ b,a (h), resulting that generically (that is, in an open and dense subset of the parameter space) θ b,a (h c ) = lim h→+∞ θ b,a (h).…”

“…Notice that still it must be proved that 2, 3, 4, 6 and 10 are forbidden. The reader is referred to [1] where, to this end, it is used the algebraic geometric approach described in Section 3.1.…”