## 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 for difference equations 39A12 Discrete version of topics in analysis
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### 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. · Zbl 1084.39001 [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 [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 [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. [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
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