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\begin{document}

\title{On the dynamics of the singularly perturbed Logistic difference equation
with two different continuous arguments}



\author[1]{A. M. A EL-Sayed}%
\author[2]{Sanaa Salman}%
\author[1]{Ali Abo-Bakr}%
\affil[1]{Alexandria University Faculty of Science}%
\affil[2]{Alexandria University}%


\vspace{-1em}



  
  \date{June 15, 2024}


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\begin{abstract}
Here, we study the dynamics of the singularly perturbed logistic
difference equation with two different continuous arguments. First of
all, local stability of the fixed points is investigated by analyzing
the corresponding characteristic equations of the linearized equations.
Secondly, we illustrate that the considered system exhibits Hopf
bifurcation. A discretized analogue of the original system is obtained
using the method of steps. Local stability and bifurcation analysis of
the discretized system are investigated. Explicit conditions for the
occurrence of a variety of complex dynamics such as fold and
Neimark-Sacker bifurcations are reached. We compare the results with
those of the associated difference equation with continuous argument
when the perturbation parameter \$\textbackslash{}epsilon
\textbackslash{}longrightarrow 0\$ and with those of the logistic delay
differential equation with two different delays when
\$\textbackslash{}epsilon \textbackslash{}longrightarrow 1\$. Finally,
numerical simulations including Lyapunov exponent, bifurcation diagrams
and phase portraits are carried out to confirm the theoretical analysis
obtained and to illustrate more complex dynamics of the system.%
\end{abstract}%



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