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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.

39A30 Stability theory for difference equations
39A20 Multiplicative and other generalized difference equations
Full Text: DOI
[1] doi:10.1080/10236190410001728104 · Zbl 1070.39025
[4] doi:10.1080/10236190701264735 · Zbl 1129.39003
[6] doi:10.1088/1751-8113/41/28/285205 · Zbl 1151.39017
[9] doi:10.1080/10236190601008737 · Zbl 1112.39012
[11] doi:10.1016/j.jmaa.2006.02.095 · Zbl 1114.39004
[12] doi:10.1080/10236190410001667977 · Zbl 1053.39008
[13] doi:10.1142/S0218127406015027 · Zbl 1141.37310
[14] doi:10.1007/s006050170042 · Zbl 1036.11002
[15] doi:10.1080/10236190600761575 · Zbl 1103.39004
[16] doi:10.1088/0305-4470/39/5/009 · Zbl 1092.39003
[17] doi:10.1155/ADE.2005.227 · Zbl 1100.39003
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