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
Publication Date: March 06, 2019 Submission Date: November 19, 2018 Revision Date: February 14, 2019 Accteptance Date: February 20, 2019
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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 $a<0$.

Keywords

• Fixed points
• critical values
• singular values
• bifurcation
• chaos
• Lyapunov exponents

•  30D05
•  37C25
•  37M25
•  58K05

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