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Global behaviour and periodicities of some fractional recursive sequences

Abdul Khaliq|E. M. Elsayed
장전수학회 (Jangjeon Mathematical Society)|2017.07|Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집) (Proceedings of the Jangjeon Mathematical Society)|제20권 제3호|pp. 421-442|DOI 10.17777/pjms.2017.20.3.011

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키워드

stabilitydifference equationsrecursive sequencesperiodic solution
주제분류수학

초록

In this paper, we obtain the expression and form of the solutions of the following recursive sequences xn+1 = xn−2xn−7 / xn−4(±1 ± xn−2xn−7) , n = 0, 1, ..., where the initial conditions x−7, x−6, x−5, x−4, x−3, x−2, x−1, x0 are arbitrary positive real numbers. We studied the equilibrium points of the given equation. Some qualitative properties such as the global stability, and the periodic character of the solutions in each case have been studied. We found expression and form of solution and presented some numerical examples by using random initial values in each case. Some figures have been given to explain the behavior of the obtained solutions by using MATLAB 6.5 to confirm the obtained results.

Abstract

In this paper, we obtain the expression and form of the solutions of the following recursive sequences xn+1 = xn−2xn−7 / xn−4(±1 ± xn−2xn−7) , n = 0, 1, ..., where the initial conditions x−7, x−6, x−5, x−4, x−3, x−2, x−1, x0 are arbitrary positive real numbers. We studied the equilibrium points of the given equation. Some qualitative properties such as the global stability, and the periodic character of the solutions in each case have been studied. We found expression and form of solution and presented some numerical examples by using random initial values in each case. Some figures have been given to explain the behavior of the obtained solutions by using MATLAB 6.5 to confirm the obtained results.

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The Dynamics of Solutions to the Equation xn+1=p+xn-k/q+xn+xn-k/xnXiaona Xu, Yongjin Li경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2010 · 제50권 제1호The Dynamics and Global Attractivity of a Rational Difference EquationAbdul Khaliq, E. M. Elsayed장전수학회 · Advanced Studies in Contemporary Mathematics2016 · 제26권 제1호Global behavior of a second-order system of difference equationsDURHASAN TURGUT TOLLU, IBRAHIM YALCINKAYA장전수학회 · Advanced Studies in Contemporary Mathematics2016 · 제26권 제4호The expressions of solutions and periodicity for some nonlinear systems of rational di¤erence equationsE. M. Elsayed장전수학회 · Advanced Studies in Contemporary Mathematics2015 · 제25권 제3호Dynamics of Recursive Sequence of Order TwoElsayed Mohammed Elsayed경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2010 · 제50권 제4호On the Solution of Fourth-Order Rational Recursive SequenceE. M. Elsayed, R.P. Agarwal장전수학회 · Advanced Studies in Contemporary Mathematics2010 · 제20권 제4호

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