Bajo, Ignacio Invariant quadrics and orbits for a family of rational systems of difference equations. (English) Zbl 1302.39023 Linear Algebra Appl. 449, 500-511 (2014). Summary: We study the existence of invariant quadrics for a class of systems of difference equations in \(\mathbb R^n\) defined by linear fractionals sharing denominator. Such systems can be described in terms of some square matrix \(A\) and we prove that there is a correspondence between non-degenerate invariant quadrics and solutions to a certain matrix equation involving \(A\). We show that if \(A\) is semisimple and the corresponding system admits non-degenerate quadrics, then every orbit of the dynamical system is contained either in an invariant affine variety or in an invariant quadric. Cited in 2 Documents MSC: 39A22 Growth, boundedness, comparison of solutions to difference equations 15A69 Multilinear algebra, tensor calculus 15A18 Eigenvalues, singular values, and eigenvectors 15A63 Quadratic and bilinear forms, inner products Keywords:quadric; difference equation; invariant set PDFBibTeX XMLCite \textit{I. Bajo}, Linear Algebra Appl. 449, 500--511 (2014; Zbl 1302.39023) Full Text: DOI arXiv References: [1] AlSharawi, Z.; Rhouma, M., Coexistence and extinction in a competitive exclusion Leslie/Gower model with harvesting and stocking, J. Difference Equ. Appl., 15, 11-12, 1031-1053 (2009) · Zbl 1176.92050 [2] Bajo, I.; Franco, D.; Perán, J., Dynamics of a rational system of difference equations in the plane, Adv. Difference Equ. (2011), Article ID 958602, 17 pp · Zbl 1232.39013 [3] Bajo, I.; Liz, E., Periodicity on discrete dynamical systems generated by a class of rational mappings, J. Difference Equ. Appl., 12, 12, 1201-1212 (2006) · Zbl 1116.39008 [4] Bajo, I.; Liz, E., Global behaviour of a second-order nonlinear difference equation, J. Difference Equ. Appl., 17, 10, 1471-1486 (2011) · Zbl 1232.39014 [5] Elsayed, E. M., Solutions of rational difference systems of order two, Math. Comput. Modelling, 55, 378-384 (2012) · Zbl 1255.39003 [6] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1994), Cambridge University Press: Cambridge University Press New York · Zbl 0801.15001 [7] Laub, A. J., Matrix Analysis for Scientists and Engineers (2005), Soc. Ind. Appl. Math.: Soc. Ind. Appl. Math. Philadelphia · Zbl 1077.15001 [8] Stević, S.; Diblík, J.; Iričanin, B.; Šmarda, Z., On some solvable difference equations and systems of difference equations, Abstr. Appl. Anal. (2012), Article ID 541761, 11 pp · Zbl 1253.39001 [9] Vinberg, E. B., A Course in Algebra (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence · Zbl 1016.00003 [10] Wang, C.-Y.; Wang, S.; Wang, W., Global asymptotic stability of equilibrium point for a family of rational difference equations, Appl. Math. Lett., 24, 714-718 (2011) · Zbl 1221.39026 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.