Bifurcation and periodically semicycles for fractional difference equation of fifth order

Volume 11, Issue 3, pp 375--382 http://dx.doi.org/10.22436/jnsa.011.03.06
Publication Date: February 14, 2018 Submission Date: October 15, 2017 Revision Date: November 07, 2017 Accteptance Date: January 10, 2018

Authors

Tarek F. Ibrahim - Mathematics Department, College of Sciences and Arts for Girls in sarat Abida, King Khalid University, Saudi Arabia. - Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt.


Abstract

Our paper takes into account a new bifurcation case of the cycle length and a fifth-order difference equation dynamics of \[ y_{m+1}=\frac{y_{m} y_{m-2}^\alpha y_{m-4}^\beta+y_{m} +y_{m-2}^\alpha +y_{m-4}^\beta + \gamma }{y_{m}y_{m-2}^\alpha + y_{m-2}^\alpha y_{m-4}^\beta+y_{m} y_{m-4}^\beta+ \gamma +1} , \quad m=0,1,2,3, \ldots, \] where \(\gamma \in [0, \infty )\) , \(\alpha,\beta\in \mathbb{Z^+} \), and \(y_{-4},y_{-3},y_{-1},y_{-2},y_0 \in (0, \; \infty )\) is took into consideration. The disturbance of initials lead to a distinction of cycle length principle of the non-trivial solutions of the equation. The principle of the track solutions structure for this equation is given. The consecutive periods of negative and positive semicycles of non-trivial solutions of this equation take place periodically with only prime period fifteen and in a period with the principles represented by either \(\{3^+,1^-, 2^+, 2^-, 1^+,1^-,1^+, 4^-\}\) or \(\{3^-,1^+, 2^-, 2^+, 1^-,1^+,1^-, 4^+\}\). From this rubric we will establish that the positive fixed point has global asymptotic stability.


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ISRP Style

Tarek F. Ibrahim, Bifurcation and periodically semicycles for fractional difference equation of fifth order, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 3, 375--382

AMA Style

Ibrahim Tarek F., Bifurcation and periodically semicycles for fractional difference equation of fifth order. J. Nonlinear Sci. Appl. (2018); 11(3):375--382

Chicago/Turabian Style

Ibrahim, Tarek F.. "Bifurcation and periodically semicycles for fractional difference equation of fifth order." Journal of Nonlinear Sciences and Applications, 11, no. 3 (2018): 375--382


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