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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

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

${u}_{n+2}^{\left(j\right)}=\frac{a+{u}_{n+1}^{\left(j+1\right)}}{{u}_{n}^{\left(j+2\right)}},\phantom{\rule{1.em}{0ex}}1\le j\le q,$

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 $\left(1,2,\cdots ,q\right)\to \left(2,3,\cdots ,q,1\right)$ and $1\le j\le q$):

The systems of $q$ Lyness’ type difference equations of order two given by ${u}_{n+2}^{\left(j\right)}{u}_{n}^{\left({\sigma }^{2}\left(j\right)\right)}={f}_{r}\left({u}_{n+1}^{\left(\sigma \left(j\right)\right)}\right)$, where ${f}_{r}$ are appropriate rational maps (11 cases) of the real line;

the system given by ${u}_{n+2}^{\left(j\right)}+{u}_{n}^{\left({\sigma }^{2}\left(j\right)\right)}={f}_{12}\left({u}_{n+1}^{\left(\sigma \left(j\right)\right)}\right)$, with ${f}_{12}\left(x\right)=\frac{\beta x}{{x}^{2}+1}$, $0<|\beta |\le 2$;

the cycle system of $q$ Todd’s type difference equations of order three given by

${u}_{n+3}^{\left(j\right)}=\frac{a+{u}_{n+2}^{\left(\sigma \left(j\right)\right)}+{u}_{n+1}^{\left({\sigma }^{2}\left(j\right)\right)}}{{u}_{n}^{\left({\sigma }^{3}\left(j\right)\right)}};$

the general cyclic system of equations of order $k$ given by

${u}_{n+k}^{\left(j\right)}=f\left({u}_{n+k-1}^{\left(\sigma \left(j\right)\right)},{u}_{n+k-2}^{\left({\sigma }^{2}\left(j\right)\right)},\cdots ,{u}_{n}^{\left({\sigma }^{k}\left(j\right)\right)}\right)·$
##### MSC:
 37E99 Low-dimensional dynamical systems 39A10 Additive difference equations 39A20 Generalized difference equations 39A30 Stability theory (difference equations)