Recent zbMATH articles in MSC 39A20 https://www.zbmath.org/atom/cc/39A20 2022-06-24T15:10:38.853281Z Werkzeug A note on general solutions to a hyperbolic-cotangent class of systems of difference equations https://www.zbmath.org/1485.39017 2022-06-24T15:10:38.853281Z "Stević, Stevo" https://www.zbmath.org/authors/?q=ai:stevic.stevo Summary: Recently there has been some interest in difference equations and systems whose forms resemble some trigonometric formulas. One of the classes of such systems is the so-called hyperbolic-cotangent class of systems of difference equations. The corresponding two-dimensional class has two delays denoted by $$k$$ and $$l$$. So far the class has been studied for the case $$k\neq l$$, and it was shown that it is practically solvable when $$\max \{k,l\}\leq 2$$. In this note we show practical solvability of the system in the case $$k=l$$, not only for small values of $$k$$ and $$l$$, but for all $$k=l\in \mathbb{N}$$, which is the first result of such generality. Application of equilibrium points in solving difference equations and a new class of solvable nonlinear systems of difference equations https://www.zbmath.org/1485.39018 2022-06-24T15:10:38.853281Z "Stević, Stevo" https://www.zbmath.org/authors/?q=ai:stevic.stevo Summary: There are not so many classes of difference equations and systems which are solvable in closed form. So it is of some interest to find new methods and ideas which can be used in solving some of them. Here we show how equilibrium points can help in finding closed-form formulas for solutions to some difference equations and systems. We demonstrate the method for the case of a class of two-dimensional nonlinear systems of difference equations by showing their theoretical solvability. We also show that some of the systems are also practically solvable, by presenting closed-form formulas for their solutions or by explaining how the formulas can be obtained. Solvability of a general class of two-dimensional hyperbolic-cotangent-type systems of difference equations https://www.zbmath.org/1485.39019 2022-06-24T15:10:38.853281Z "Stević, Stevo" https://www.zbmath.org/authors/?q=ai:stevic.stevo Summary: We show that the following class of two-dimensional hyperbolic-cotangent-type systems of difference equations $x_{n+1}=\frac{u_{n-k}v_{n-l}+a}{u_{n-k}+v_{n-l}},\qquad y_{n+1}= \frac{w_{n-k}s_{n-l}+a}{w_{n-k}+s_{n-l}},\quad n\in{\mathbb{N}}_0,$ where $$k,l\in{\mathbb{N}}_0, a\in{\mathbb{C}}, u_{-j}, w_{-j}\in{\mathbb{C}}, j=\overline{1,k}, v_{-j'}, s_{-j'}, j'=\overline{1,l}$$, and each of the sequences $$u_n, v_n, w_n, s_n$$ is equal to $$x_n$$ or $$y_n$$, is theoretically solvable. When $$k=0$$ and $$l=1$$, we show that the systems are practically solvable by presenting closed-form formulas for their solutions. To do this, we employ a constructive method, which is possible to use on the complex domain, presenting in this way a new and elegant solution to the problem in this case, and giving a hint how such type of systems can be solved. Form of the periodic solutions of some systems of higher order difference equations https://www.zbmath.org/1485.39023 2022-06-24T15:10:38.853281Z "Göcen, Melih" https://www.zbmath.org/authors/?q=ai:gocen.melih "Cebeci, Adem" https://www.zbmath.org/authors/?q=ai:cebeci.adem Behaviour of orbits and periods of a two dimensional dynamical system associated to a special QRT-map https://www.zbmath.org/1485.39029 2022-06-24T15:10:38.853281Z "Bastien, G." https://www.zbmath.org/authors/?q=ai:bastien.guy "Rogalski, M." https://www.zbmath.org/authors/?q=ai:rogalski.marc The authors present a complete study of the orbits and the period of a particular QRT-map, namely \left\{ \begin{aligned} X x &= \frac{ey^{2}+gy+h}{y^{2}+by+d}, \\ Y y &= \frac{dX^{2}+fX+h}{X^{2}+cX+e}, \end{aligned} \right.\tag{1} for some specific value of the parameters. The map is constructed with the recipe introduced in [\textit{G. R. W. Quispel} et al., Phys. Lett., A 126, No. 7, 419--421 (1988; Zbl 0679.58023)] and is integrable in the sense that it admits an invariant bi-quadratic curve and preserves a measure on the plane. This paper contains a complete study of the dynamical properties of the system (1), including a construction of the representation of the map as a rotation on the unit circle, the limit of the rotation number near the critical values, and finally the possible periods. In this sense the paper is part of a series of papers on special QRT-maps by the same authors and their collaborators (see [\textit{G. Bastien} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 23, No. 4, Article ID 1350071, 18 p. (2013; Zbl 1270.39009); \textit{G. Bastien} and \textit{M. Rogalski}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 22, No. 11, Paper No. 1250266, 10 p. (2012; Zbl 1258.37044); Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 20, No. 6, 727--745 (2013; Zbl 1282.39014); in: Difference equations, discrete dynamical systems and applications, ICDEA, Barcelona, Spain, July 23--27, 2012. Proceedings of the 18th international conference. Berlin: Springer. 237--265 (2016; Zbl 1355.39014)]). The paper is rather technical, but several interesting ideas and concepts are worth of interest. Indeed, the dynamical aspects are often overlooked in the literature on integrable systems, which is often more focused on the algebro-geometric point of view, see, e.g., the book [\textit{J. J. Duistermaat}, Discrete integrable systems. QRT maps and elliptic surfaces. Berlin: Springer (2010; Zbl 1219.14001)] and all the development by the Japanese school of integrable systems. Particularly interesting is the computation of the limits of the rotational numbers, including results on the continuity of these numbers. This kind of approach is remarkable and might be useful to analyse and maybe even sieve other integrable systems. Reviewer: Giorgio Gubbiotti (Milano)