In this paper we study the partial differential equation with piecewise constant argument of the form : \[ \begin{array}{lll} x_t(t,s)=&A(t)x(t,s)+B(t,s)x([t],s)+C(t,s)x(t,[s])+\\[0.5cm] &D(t,s)x([t],[s])+f(x(t,[s])),\ \ t,s\in \R^{+}=(0,\infty) \end{array} \] where $A(t)$ is a $k\times k$ invertible and continuous matrix function on $\R^{+}$, $B(t,s)$, $C(t,s)$, $D(t,s)$ are $k \times k$ continuous and bounded matrix functions on $\R^{+}\times \R^{+}$, $[t]$, $[s]$ are the integral parts of $t,s$ respectively and $f:\R^k\rightarrow \R^k$ is a continuous function. More precisely under some conditions on the matrices $A(t)$, $B(t,s)$, $C(t,s)$, $D(t,s)$ and the function $f$ we investigate the asymptotic behaviour of the solutions of the above equation. \end{abstract}

In this paper, we study the conditions under which the following symmetric system of difference equations with exponential terms: \[ x_{n+1} =a_1\frac{y_n}{b_1+y_n} +c_1\frac{x_ne^{k_1-d_1x_n}}{1+e^{k_1-d_1x_n}},\] \[ y_{n+1} =a_2\frac{x_n}{b_2+x_n} +c_2\frac{y_ne^{k_2-d_2y_n}}{1+e^{k_2-d_2y_n}}\] where $a_i$, $b_i$, $c_i$, $d_i$, $k_i$, for $i=1,2$, are real constants and the initial values $x_0$, $y_0$ are real numbers, undergoes Neimark-Sacker, flip and transcritical bifurcation. The analysis is conducted applying center manifold theory and the normal form bifurcation analysis.

In this paper, we study the stability of the zero equilibrium and the occurrence of flip bifurcation on the following system of difference equations: \[x_{n+1} =a_1\frac{y_n}{b_1+y_n} +c_1\frac{x_ne^{k_1-d_1x_n}}{1+e^{k_1-d_1x_n}},\]\\ \[y_{n+1} =a_2\frac{z_n}{b_2+z_n} +c_2\frac{y_ne^{k_2-d_2y_n}}{1+e^{k_2-d_2y_n}},\]\\ \[z_{n+1} =a_3\frac{x_n}{b_3+x_n} +c_3\frac{z_ne^{k_3-d_3z_n}}{1+e^{k_3-d_3z_n}}\] where $a_i$, $b_i$, $c_i$, $d_i$, $k_i$, for $i=1,2,3$, are real constants and the initial values $x_0$, $y_0$ and $z_0$ are real numbers. We study the stability of this system in the special case when one of the eigenvalues is equal to -1 and the remaining eigenvalues have absolute value less than 1, using center manifold theory.