zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Stability of nonlinear autonomous quadratic discrete systems in the critical case. (English) Zbl 1200.39005
Summary: Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalue $\lambda =1$ of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

MSC:
 39A30 Stability theory (difference equations) 39A12 Discrete version of topics in analysis
Full Text:
References:
 [1] R. P. Agarwal, Difference Equations and Inequalities, Theory, Methods and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. · Zbl 0952.39001 [2] R. P. Agarwal, M. Bohner, S. R. Grace, and D. O’Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, 2005. · doi:10.1155/9789775945198 [3] N. G. Chetaev, Dynamic Stability, Nauka, Moscow, Russia, 1965. [4] S. N. Elaydi, An Introduction to Difference Equations, Springer, London, UK, 3rd edition, 2005. · Zbl 1071.39001 [5] A. Halanay and V. R\uasvan, Stability and Stable Oscillations in Discrete Time Systems, Gordon and Breach Science, Taipei, Taiwan, 2002. [6] V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, vol. 251 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2002. · Zbl 1014.39001 [7] D. I. Martynjuk, Lectures on the Qualitative Theory of Difference Equations, “Naukova Dumka”, Kiev, Ukraine, 1972. [8] V. E. Slyusarchuk, “Essentially unstable solutions of difference equations,” Ukrainian Mathematical Journal, vol. 51, no. 12, pp. 1659-1672, 1999 (Russian), translation in Ukrainian Mathematical Journal, vol. 51, no. 12, pp. 1875-1891, 1999. · Zbl 0937.39008 · doi:10.1007/BF02525136 [9] V. E. Slyusarchuk, “Essentially unstable solutions of difference equations in a Banach space,” Differentsial’nye Uravneniya, vol. 35, no. 7, pp. 982-989, 1999 (Russian), translation in Differential Equations, vol. 35, no. 7, pp. 992-999, 1999. · Zbl 0968.39006 [10] V. E. Slyusarchuk, “Theorems on the instability of systems with respect to linear approximation,” Ukrains’kyi Matematychnyi Zhurnal, vol. 48, no. 8, pp. 1104-1113, 1996 (Russian), translation in Ukrainian Mathematical Journal, vol. 48, no. 8, pp. 1251-1262, 1996. · Zbl 0941.34061 · doi:10.1007/BF02383871 [11] J. Diblík, D. Ya. Khusainov, and I. V. Grytsay, “Stability investigation of nonlinear quadratic discrete dynamics systems in the critical case,” Journal of Physics: Conference Series, vol. 96, no. 1, Article ID 012042, 2008. · doi:10.1088/1742-6596/96/1/012042 [12] F. P. Gantmacher, The Theory of Matrices, vol. I, AMS Chelsea Publishing, Providence, RI, USA, 2002. · Zbl 1002.74002 [13] I. G. Malkin, Teoriya Ustoichivosti Dvizheniya, Nauka, Moscow, Russia, 2nd edition, 1966. · Zbl 0136.08502