Closed-Form Solutions of a Nonlinear Rational Second-Order Three-Dimensional System of Difference Equations

Messaoud Berkal, Taha Radwan (), Mehmet Gümüş, Raafat Abo-Zeid and Karim K. Ahmed
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Messaoud Berkal: Department of Applied Mathematics, Abdelhafid Boussouf University Center, Mila, R.P 26, Mila 43000, Algeria
Taha Radwan: Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia
Mehmet Gümüş: Department of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University, 67100 Zonguldak, Türkiye
Raafat Abo-Zeid: Department of Basic Science, The Higher Institute for Engineering & Technology, Al-Obour, Cairo 10587, Egypt
Karim K. Ahmed: Department of Mathematics, Faculty of Engineering, German International University (GIU), New Administrative Capital, Cairo 11835, Egypt

Mathematics, 2025, vol. 13, issue 20, 1-17

Abstract: In this paper, we investigate the behavior of solutions to a nonlinear system of rational difference equations of order two, defined by x n + 1 = x n y n − 1 y n ( a + b x n y n − 1 ) , y n + 1 = y n z n − 1 z n ( c + d y n z n − 1 ) , z n + 1 = z n x n − 1 x n ( e + f z n x n − 1 ) , where n denotes a nonzero integer; the parameters a , b , c , d , e , f are real constants; and the initial conditions x − 1 , x 0 , y − 1 , y 0 , z − 1 , z 0 are nonzero real numbers. By applying a suitable variable transformation, we reduce the original coupled system to three independent rational difference equations. This allows for separate analysis using established methods for second-order nonlinear recurrence relations. We derive explicit solutions and examine the qualitative behavior, including boundedness and periodicity, under different conditions. Our findings contribute to the theory of rational difference equations and offer insights for higher-order systems in applied sciences.

Keywords: system of difference equations; solution; nonlinear systems; rational difference equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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