On Schur Stability and Oscillation of Linear Difference Equation Systems with Constant Coefficients

Year 2024, Volume: 12 Issue: 2, 124 - 129, 28.10.2024

Ahmet Duman , Ramazan Çakıroğlu , Kemal Aydın

Abstract

In this study, the solutions of linear difference equation systems with constant coefficients were examined with respect to whether they were Schur stability and oscillatory or not. A new parameter $\gamma (A)$ which indicates the quality of Schur stability and oscillatory of the system has been defined. For Schur stable and oscillation linear difference equation system with constant coefficients, continuity theorems which show how much change is permissible without disturbing the Schur stability and oscillatory have been proved, and some examples illustrating the efficiency of the theorems have been given.

Keywords

Difference equation systems, Schur stability, oscillation, sensitivity, perturbation systems, condition number

References

• [1] R. C¸ akıro˘glu, A. Duman and K. Aydın, On Schur stability and oscillation of second order difference equations, J. Inst. Sci. and Tech., 11(4) (2021), 3098-31110, doi:10.21597/jist.902856.
• [2] S. Elaydi, An Introduction to Difference Equations, Springer, New York, 2005.
• [3] S. Goldberg, Introduction to Difference Equations, Science Editions, New York, 1958.
• [4] O¨ . Akın, H. Bulgak, Lineer fark denklemleri ve kararlılık teorisi, Selc¸uk U¨ niversitesi Uygulamalı Matematik Aras¸tırma Merkezi Yayınları, Konya, 1998.
• [5] J. Sunday, On the Oscillation Criteria and Computation of Third Order Oscillatory Differential Equations, Communications in Mathematics and Applications, 9(4) (2018), 615-626.
• [6] O. Sinan, S. Baydak, A. Duman and K. Aydın, Computation of the Solutions of Lyapunov Matrix Equations with Iterative Decreasing Dimension Method., Fundam. J. Math. Appl.;5(2) (2022), 98-105, doi:10.33401/fujma.996668.
• [7] A. Duman and K. Aydın, Sensitivity of Schur Stability of Systems of Linear Difference Equations with Constant Coefficients, Scientific Research and Essays, 6 (28) (2011), 5846-5854.
• [8] A. Ya. Bulgakov and S. K. Godunov, Circle Dichotomy of the Matrix Spectrum, Siberia Math. J., 29(5) (1988), 59-70.
• [9] A. Ya. Bulgakov, Matrix Computations with Guaranteed Accuracy in Stability Theory, Selc¸uk U¨ niversitesi Yayınları, Konya, 1995.
• [10] K. Aydın, Periyodik Adi Diferensiyel Denklem Sistemlerinin Asimtotik Kararlılı˘gı ic¸in S¸ art Sayısı, Doktora Tezi, Konya, 1995.
• [11] H. Bulgak, Pseudoeigenvalues, Spectral Portrait of a Matrix and Their Connections with Different Criteria of Stability, Error Control and Adaptivity in Scientific Computing, (1999), 95-124
• [12] A. Duman and K. Aydın, Sensitivity of Hurwitz Stability of Linear Differential Equation Systems with Constant Coefficients, International Journal of Geometric Methods in Modern Physics, 14(6) (2017), 1750084.