Bastien, G.; Rogalski, M. Global behaviour of solutions of cyclic systems of \(q\) order 2 or 3 generalized Lyness’ difference equations and of other more general equations of higher order. (English) Zbl 1247.37037 J. Difference Equ. Appl. 17, No. 11, 1651-1672 (2011). The authors consider several cyclic systems of difference equations of order \(2\) or \(3\) or higher order and analyse the global behaviour of their solutions. For instance, they study the system of \(q\) difference equations of order 2 given by \[ u_{n+2}^{(j)}=\frac{ a+u_{n+1}^{(j+1)} } {u_{n}^{(j+2)}},\quad 1\leq j\leq q, \] where \(a\) is a positive constant. By using the method of geometric unfolding of a difference equation, that is, by dealing with the associated discrete dynamical system, the authors obtain information about global periodicity (the case \(a=1\) which was already considered in [B. Iričanin and S. Stević, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 13, No. 3–4, 499–507 (2006; Zbl 1098.39003)] but using direct – and complicated – calculations) and other dynamical questions such as the localization of equilibrium points, and the global behaviour of the solutions lying in some invariant sets.With the same strategy of the geometric unfolding, other systems considered in the paper are (here, \(\sigma\) is the cyclic permutation \((1,2,\dots,q) \rightarrow (2,3,\dots,q,1)\) and \(1\leq j\leq q\)): (1) The systems of \(q\) Lyness’ type difference equations of order two given by \(u_{n+2}^{(j)}u_{n}^{(\sigma^{2}(j))}=f_{r}(u_{n+1}^{(\sigma(j))})\), where \(f_r\) are appropriate rational maps (\(11\) cases) of the real line; (2) the system given by \(u_{n+2}^{(j)}+u_{n}^{(\sigma^{2}(j))}=f_{12}(u_{n+1}^{(\sigma(j))})\), with \(f_{12}(x)=\frac{\beta x}{x^2+1}\), \(0<|\beta|\leq 2\); (3) the cycle system of \(q\) Todd’s type difference equations of order three given by \[ u_{n+3}^{(j)}=\frac{a+u_{n+2}^{(\sigma(j))}+u_{n+1}^{(\sigma^{2}(j))} } {u_{n}^{(\sigma^{3}(j))}}; \] (4) the general cyclic system of equations of order \(k\) given by \[ u_{n+k}^{(j)}=f(u_{n+k-1}^{(\sigma(j))}, u_{n+k-2}^{(\sigma^{2}(j))} ,\dots, u_{n}^{(\sigma^{k}(j))}). \] Reviewer: Antonio Linero Bas (Murcia) Cited in 3 Documents MSC: 37E99 Low-dimensional dynamical systems 39A10 Additive difference equations 39A20 Multiplicative and other generalized difference equations 39A30 Stability theory for difference equations Keywords:dynamical systems; difference equations; Lyness’ equations; periods; Todd’s equations; global periodicity; global behaviour; invariant sets; geometric unfolding Citations:Zbl 1098.39003 PDFBibTeX XMLCite \textit{G. Bastien} and \textit{M. Rogalski}, J. Difference Equ. Appl. 17, No. 11, 1651--1672 (2011; Zbl 1247.37037) Full Text: DOI References: [1] DOI: 10.1080/10236190410001728104 · Zbl 1070.39025 · doi:10.1080/10236190410001728104 [2] DOI: 10.1016/j.jmaa.2004.06.035 · Zbl 1070.39024 · doi:10.1016/j.jmaa.2004.06.035 [3] DOI: 10.1155/ADE.2005.227 · Zbl 1100.39003 · doi:10.1155/ADE.2005.227 [4] DOI: 10.1016/j.jmaa.2006.02.095 · Zbl 1114.39004 · doi:10.1016/j.jmaa.2006.02.095 [5] DOI: 10.1080/10236190601008737 · Zbl 1112.39012 · doi:10.1080/10236190601008737 [6] Bastien G., Adv. Difference Equ. (2009) [7] DOI: 10.1006/jdeq.1997.3359 · Zbl 0944.37026 · doi:10.1006/jdeq.1997.3359 [8] Bourbaki N., Topologie Générale, Chapitre 7: les groupes additives \(\mathbb{R}\)n (1963) [9] DOI: 10.1080/10236190701264735 · Zbl 1129.39003 · doi:10.1080/10236190701264735 [10] A. Cima, A. Gasull, and V. Mañosa, On two and three periodic Lyness difference equation, arXiv: 09125031v1 [math. DS], 26 December, 2009 [11] Iricanin B., Dyn. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal. 13 pp 499– (2006) [12] Kocic V.L., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993) · Zbl 0787.39001 · doi:10.1007/978-94-017-1703-8 [13] Papaschinopoulos G., Adv. Difference Equ. (2007) [14] Yalçinkaya I., Adv. Difference Equ. (2008) [15] Zeeman E.C., Geometric Unfolding of a Difference Equation (1996) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.