Chaotic behavior in real dynamics and singular values of family of generalized generating function of Apostol-Genocchi numbers

Volume 19, Issue 1, pp 41--50 http://dx.doi.org/10.22436/jmcs.019.01.06
Publication Date: March 06, 2019 Submission Date: November 19, 2018 Revision Date: February 14, 2019 Accteptance Date: February 20, 2019

Authors

Mohammad Sajid - College of Engineering, Qassim University, Buraidah, Al-Qassim, Saudi Arabia.


Abstract

Chaotic behavior in the real dynamics and singular values of a two-parameter family of generalized generating function of Apostol-Genocchi numbers, \(f_{\lambda,a}(z)=\lambda \frac{2z}{e^{az}+1}\), \(\lambda, a\in \mathbb{R} \backslash \{0\}\), are investigated. The real fixed points of \(f_{\lambda,a}(z)\) and their nature are studied. It is seen that bifurcation and chaos occur in the real dynamics of \(f_{\lambda,a}(z)\). It is also found that the function \(f_{\lambda,a}(z)\) has infinitely many singular values for \(a>0\) and \(a<0\). The critical values of \(f_{\lambda,a}(z)\) lie inside the open disk, the annulus and exterior of the open disk at center origin for \(a>0\) and \(a<0\).


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