Fixed points and dynamics on generating function of Genocchi numbers

Volume 9, Issue 3, pp 933--939 http://dx.doi.org/10.22436/jnsa.009.03.22

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1587 Downloads
• 2595 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

Keywords

• Fixed point
• Genocchi number
• chaos.

MSC

•  30D05
•  37C25
•  37F45
•  58K05

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)