Geodesic
Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms
Kulenovic, M. R. S.1 ; Moranjkic, S.2 ; Nurkanovic, Z.2
1 Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA
2 Department of Mathematics, University of Tuzla, 75350 Tuzla, Bosnia and Herzegovina
Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 7, p. 3477-3489.

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

We investigate the global asymptotic stability and Naimark-Sacker bifurcation of the difference equation
$x_{n+1} =\frac{F}{bx_nx_{n-1} + cx^2_{n-1} + f} , n = 0, 1, ... ,$
with non-negative parameters and nonnegative initial conditions $x_{-1}, x_0$ such that $bx_0x_{-1} + cx^2_{-1} + f > 0$. By using fixed point theorem for monotone maps we find the region of parameters where the unique equilibrium is globally asymptotically stable.
DOI : 10.22436/jnsa.010.07.11
Classification : 39A10, 39A28, 39A30
Keywords: Attractivity, bifurcation, difference equation, invariant, Naimark-Sacker bifurcation, periodic solution.

Kulenovic, M. R. S. 1 ; Moranjkic, S. 2 ; Nurkanovic, Z. 2

1 Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA
2 Department of Mathematics, University of Tuzla, 75350 Tuzla, Bosnia and Herzegovina 
@article{JNSA_2017_10_7_a10,
     author = {Kulenovic, M. R. S. and Moranjkic, S. and Nurkanovic, Z.},
     title = {Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {3477-3489},
     publisher = {mathdoc},
     volume = {10},
     number = {7},
     year = {2017},
     doi = {10.22436/jnsa.010.07.11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.07.11/}
}
TY  - JOUR
AU  - Kulenovic, M. R. S.
AU  - Moranjkic, S.
AU  - Nurkanovic, Z.
TI  - Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms
JO  - Journal of nonlinear sciences and its applications
PY  - 2017
SP  - 3477
EP  - 3489
VL  - 10
IS  - 7
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.07.11/
DO  - 10.22436/jnsa.010.07.11
LA  - en
ID  - JNSA_2017_10_7_a10
ER  - 
%0 Journal Article
%A Kulenovic, M. R. S.
%A Moranjkic, S.
%A Nurkanovic, Z.
%T Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms
%J Journal of nonlinear sciences and its applications
%D 2017
%P 3477-3489
%V 10
%N 7
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.07.11/
%R 10.22436/jnsa.010.07.11
%G en
%F JNSA_2017_10_7_a10
Kulenovic, M. R. S.; Moranjkic, S.; Nurkanovic, Z. Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms. Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 7, p. 3477-3489. doi : 10.22436/jnsa.010.07.11. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.07.11/

[1] Amleh, A. M.; Camouzis, E.; Ladas, G. On the dynamics of a rational difference equation, I, Int. J. Difference Equ., Volume 3 (2008), pp. 1-35

[2] Amleh, A. M.; Camouzis, E.; Ladas, G. On the dynamics of a rational difference equation, II, Int. J. Difference Equ., Volume 3 (2008), pp. 195-225

[3] Hale, J. K.; Kocak, H. Dynamics and bifurcations, Texts in Applied Mathematics, Springer-Verlag, New York, 1991

[4] Janowski, E. A.; Kulenović, M. R. S. Attractivity and global stability for linearizable difference equations, Comput. Math. Appl., Volume 57 (2009), pp. 1592-1607 | Zbl | DOI

[5] Kent, C. M.; Sedaghat, H. Global attractivity in a quadratic-linear rational difference equation with delay, J. Difference Equ. Appl., Volume 15 (2009), pp. 913-925 | Zbl | DOI

[6] Kent, C. M.; Sedaghat, H. Global attractivity in a rational delay difference equation with quadratic terms, J. Difference Equ. Appl., Volume 17 (2011), pp. 457-466 | Zbl | DOI

[7] Kulenović, M. R. S.; Ladas, G. Dynamics of second order rational difference equations, With open problems and conjectures, Chapman & Hall/CRC, Boca Raton, FL, 2001

[8] Kulenović, M. R. S.; Merino, O. Discrete dynamical systems and difference equations with Mathematica, Chapman & Hall/CRC, Boca Raton, FL, 2002 | DOI

[9] Kulenović, M. R. S.; Merino, O. A global attractivity result for maps with invariant boxes, Discrete Contin. Dyn. Syst. Ser. B, Volume 6 (2006), pp. 97-110 | Zbl

[10] Kulenović, M. R. S.; Merino, O. Global bifurcation for discrete competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, Volume 12 (2009), pp. 133-149 | Zbl

[11] Kulenović, M. R. S.; Merino, O. Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Volume 20 (2010), pp. 2471-2486 | Zbl | DOI

[12] Kulenović, M. R. S.; Pilav, E.; Silić, E. Naimark-Sacker bifurcation of a certain second order quadratic fractional difference equation, J. Math. Comput. Sci., Volume 4 (2014), pp. 1025-1043

[13] Kuznetsov, Y. A. Elements of applied bifurcation theory, Second edition. Applied Mathematical Sciences, Springer- Verlag, New York, 1998

[14] Robinson, C. Stability, symbolic dynamics, and chaos, Stud. Adv. Math. Boca Raton, CRC Press, FL, 1995

[15] Sedaghat, H. Global behaviours of rational difference equations of orders two and three with quadratic terms, J. Difference Equ. Appl., Volume 15 (2009), pp. 215-224 | DOI | Zbl

[16] Wiggins, S. Introduction to applied nonlinear dynamical systems and chaos, Second edition, Texts in Applied Mathematics, Springer-Verlag, New York, 2003

Cité par Sources :