Fixed points and dynamics on generating function of Genocchi numbers
-
1673
Downloads
-
2852
Views
Authors
Dongkyu Lim
- Department of Mathematics, Kyungpook National University, Daegu, 41566, S. Korea.
Abstract
Recently, there have been many works related with dynamics of various functions. In this paper, singular
values and fixed points of generating function of Genocchi numbers, \(g_\lambda(z) = \lambda \frac{2z}
{e^z+1}, (\lambda\in R) > 1\), are
investigated. It is shown that the function \(g_\lambda(z)\) has infinitely many singular values and its critical values
lie in the left half plane and one point on the real axis in the right half plane. Further, the real fixed
points of \(g_\lambda(z)\) and their nature are determined. Finally, we provide numerical evidence of the existence of
chaotic phenomena by illustrating bifurcation diagrams of system and by calculating the Lyapunov exponent.
Share and Cite
ISRP Style
Dongkyu Lim, Fixed points and dynamics on generating function of Genocchi numbers, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 3, 933--939
AMA Style
Lim Dongkyu, Fixed points and dynamics on generating function of Genocchi numbers. J. Nonlinear Sci. Appl. (2016); 9(3):933--939
Chicago/Turabian Style
Lim, Dongkyu. "Fixed points and dynamics on generating function of Genocchi numbers." Journal of Nonlinear Sciences and Applications, 9, no. 3 (2016): 933--939
Keywords
- Fixed point
- Genocchi number
- chaos.
MSC
References
-
[1]
K. T. Alligood, T. D. Sauer, J. A. Yorke, Chaos: An introduction to dynamical systems, Springer, New York (1997)
-
[2]
S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Appl. Math. Comput., 233 (2014), 599-607.
-
[3]
G. P. Kapoor, M. G. P. Prasad , Dynamics of \((e^z - 1)/z\): the Julia set and bifurcation, Ergodic Theory Dynam. Systems, 18 (1998), 1363-1383.
-
[4]
T. Kim, A note on the q-Genocchi numbers and polynomials, J. Inequal. Appl., 2007 (2007), 8 pages.
-
[5]
T. Nayak, M. G. P. Prasad, Iteration of certain meromorphic functions with unbounded singular values, Ergodic Theory Dynam. Systems, 30 (2010), 877-891.
-
[6]
M. G. P. Prasad, Chaotic burst in the dynamics of \(f_\lambda(z) = \lambda\frac{sinh(z)}{ z}\), Regul. Chaotic Dyn., 10 (2005), 71-80.
-
[7]
M. Sajid, Singular values and fixed points of family of generating function of Bernoulli's numbers , J. Nonlinear Sci. Appl., 8 (2015), 17-22.
-
[8]
M. Sajid, A. S. Alsuwaiyan, Chaotic behavior in the real dynamics of a one parameter family of functions, Int. J. Appl. Sci. Eng., 12 (2014), 289-301.
-
[9]
S. H. Strogatz, Nonlinear dynamics and chaos, Addison-Wesley Publishing Company, New York (1994)