Three-dimensional max-type cyclic system of difference equations

Since the discrete structure underlying quantum gravity leads to ‘difference equations’ as the basic dynamical laws, new problems to determine properties of their solutions arise. The type of these equations is different from standard ones, and questions asked sometimes differ from what traditional investigations were interested in, which requires new tools to be developed. One issue is how to derive semi-classical properties of solutions in regimes where difference equations are close to continuum behavior. More generally, an important question is how such semi classical behavior can emerge dynamically in a quantum system. Recently, there has been a great interest in studying nonlinear difference equations since many models describing real life situations in population biology, economics, probability theory, genetics, psychology, sociology etc. are represented by these equations. As we know the investigation of max-type difference equations attracted some attention recently. Some of the max-type difference equations arises naturally in certain models in automatic control theory (Stevic, 2010), and other fields. There has been some recent interest in studying maxtype difference equations, see, for instance, (Berenhaut et al., 2006; Cinar et al., 2005; Grove and Ladas, 2005; Iricanin and Stevic, 2009; Mishev et al., 2003) and references cited therein. These papers study mostly some particular cases of the following max-type difference equations