Geodesic
Bifurcation and periodically semicycles for fractional difference equation of fifth order
Ibrahim, Tarek F. 1
1 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
Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 3, p. 375-382.

Voir la notice de l'article provenant de la source International Scientific Research Publications

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.
DOI : 10.22436/jnsa.011.03.06
Classification : 39A10, 39A28
Keywords: Semicycles, solutions, difference equations, oscillatory solution, global stability

Ibrahim, Tarek F.  1

1 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 
@article{JNSA_2018_11_3_a5,
     author = {Ibrahim, Tarek F. },
     title = {Bifurcation and periodically semicycles for fractional difference equation of fifth order},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {375-382},
     publisher = {mathdoc},
     volume = {11},
     number = {3},
     year = {2018},
     doi = {10.22436/jnsa.011.03.06},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.03.06/}
}
TY  - JOUR
AU  - Ibrahim, Tarek F. 
TI  - Bifurcation and periodically semicycles for fractional difference equation of fifth order
JO  - Journal of nonlinear sciences and its applications
PY  - 2018
SP  - 375
EP  - 382
VL  - 11
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.03.06/
DO  - 10.22436/jnsa.011.03.06
LA  - en
ID  - JNSA_2018_11_3_a5
ER  - 
%0 Journal Article
%A Ibrahim, Tarek F. 
%T Bifurcation and periodically semicycles for fractional difference equation of fifth order
%J Journal of nonlinear sciences and its applications
%D 2018
%P 375-382
%V 11
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.03.06/
%R 10.22436/jnsa.011.03.06
%G en
%F JNSA_2018_11_3_a5
Ibrahim, Tarek F. . Bifurcation and periodically semicycles for fractional difference equation of fifth order. Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 3, p. 375-382. doi : 10.22436/jnsa.011.03.06. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.03.06/

[1] A. M. Ahmed On the dynamics of a higher-order rational difference equation, Discrete Dyn. Nat. Soc., Volume 2011 (2011), pp. 1-8

[2] Q. Din Dynamics of a discrete Lotka-Volterra model , Adv. Difference Equ., Volume 2013 (2013), pp. 1-13 | DOI

[3] Elabbasy, E. M.; Elsayed, E. M. On the global attractivity of difference equation of higher order, Carpathian J. Math., Volume 24 (2008), pp. 45-53

[4] El-Metwally, H.; Grove, E. A.; Ladas, G.; H. D. Voulov On the global attractivity and the periodic character of some difference equations, J. Differ. Equations Appl., Volume 7 (2001), pp. 837-850 | Zbl | DOI

[5] Elsayed, E. M.; T. F. Ibrahim Periodicity and Solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat., Volume 44 (2015), pp. 1361-1390 | DOI | Zbl

[6] Elsayed, E. M.; T. F. Ibrahim Solutions and Periodicity of a Rational Recursive Sequences of order Five, Bull. Malays. Math. Sci. Soc., Volume 38 (2015), pp. 95-112 | DOI | Zbl

[7] T. F. Ibrahim Periodicity and Global Attractivity of Difference Equation of Higher Order, J. Comput. Anal. Appl., Volume 16 (2014), pp. 552-564 | Zbl

[8] Ibrahim, T. F.; El-Moneam, M. A. Global stability of a higher-order difference equation, Iran. J. Sci. Technol. Trans. A Sci., Volume 41 (2017), pp. 51-58 | DOI

[9] Ibrahim, T. F.; N. Touafek Max-Type System Of Difference Equations With Positive Two-Periodic Sequences, Math. Methods Appl. Sci., Volume 37 (2014), pp. 2541-2553 | Zbl | DOI

[10] Kocić, V. L.; Ladas, G. Global behavior of Nonlinear Difference Equations of Higher Order with Applications, Academic Publishers Group, Dordrecht, 1993 | DOI

[11] Ladas, G. Open problems and conjectures, J. Difference Equa. Appl., Volume 10 (2004), pp. 1119-1127

[12] X. Li Global behavior for a fourth-order rational difference equation, J. Math. Anal. Appl., Volume 312 (2005), pp. 555-563 | DOI

[13] X. Li The rule of trajectory structure and global asymptotic stability for a nonlinear difference equation, Appl. Math. Lett., Volume 19 (2006), pp. 1152-1158 | DOI

[14] Touafek, N. On a second order rational difference equation , Hacet. J. Math. Stat., Volume 41 (2012), pp. 867-874

[15] Yalinkaya, I.; Cinar, C. On the positive solutions of the difference equation system \(x_{n+1} = \frac{1}{ z_n} , y_{n+1} = \frac{y_n}{ x_n+y_{n-1}} , z_{n+1} = \frac{1}{ x_{n-1}}\) , J. Inst. Math. Comp. Sci., Volume 18 (2005), pp. 135-136

[16] Zayed, E. M. E.; El-Moneam, M. A. On the global attractivity of two nonlinear difference equations, J. Math. Sci., Volume 177 (2011), pp. 487-499 | Zbl | DOI

Cité par Sources :