Stability and flip bifurcation of a three dimensional exponential system
of difference equations
Abstract
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.
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