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Results and conjectures about order $$q$$ Lyness’ difference equation $$u_{n+q}u_n=a+u_{n+q - 1}+ \dots +u_{n+1}$$ in $$\mathbb R_{\ast}^+$$, with a particular study of the case $$q=3$$. (English) Zbl 1177.39021
Summary: We study order $$q$$ Lyness’ difference equation in $$\mathbb R_{\ast }^{+}:u_{n+q}u_{n}=a+u_{n+q - 1}+ \dots +u_{n+1}$$, with $$a>0$$ and the associated dynamical system $$F_{a}$$ in $$\mathbb R_{\ast }^{+q}$$. 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 $$F_{a}^{2}$$ and state some conjectures on the associated dynamical system in $$\mathbb R_{\ast }^{+q}$$ in more general cases.

##### MSC:
 39A30 Stability theory for difference equations 39A20 Multiplicative and other generalized difference equations
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##### References:
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