Results and Conjectures about Order q Lyness' Difference Equation un+qun=a+un+q−1+⋯+un+1 in ℝ∗+, with a Particular Study of the Case q=3

We study order q Lyness' difference equation in ℝ∗+:un+qun=a+un+q−1+⋯+un+1, with a>0 and the associated dynamical system Fa in ℝ∗+q. We study its solutions (divergence, permanency, local stability of the equilibrium). We prove some results, about the first three invariant functions and the topological nature of the corresponding invariant sets, about the differential at the equilibrium, about the role of 2-periodic points when q is odd, about the nonexistence of some minimal periods, and so forth and discuss some problems, related to the search of common period to all solutions, or to the second and third invariants. We look at the case q=3 with new methods using new invariants for the map Fa2 and state some conjectures on the associated dynamical system in ℝ∗+q in more general cases.