Global dynamical properties of two discrete-time exponential systems

Abstract

We explore the global dynamics of following three directional discrete-time exponential systems of difference equations: \[{x_{n + 1}} = \frac{{{\alpha _{19}} + {\beta _{19 }}{e^{ - {y_n}}}}}{{{\gamma _{19}} + {z_{n - 1}}}},\,\quad {y_{n + 1}} = \frac{{{\alpha _{20}} + {\beta _{20 }}{e^{ - {z_n}}}}}{{{\gamma _{20}} + {x_{n - 1}}}},\quad {z_{n + 1}} = \frac{{{\alpha _{21}} + {\beta _{21}}{e^{ - {x_n}}}}}{{{\gamma _{21}} + {y_{n - 1}}}},\] \[{x_{n + 1}} = \frac{{{\alpha _{22}} + {\beta _{22 }}{e^{ - {z_n}}}}}{{{\gamma _{22}} + {x_{n - 1}}}},\quad {y_{n + 1}} = \frac{{{\alpha _{23}} + {\beta _{23}}{e^{ - {x_n}}}}}{{{\gamma _{23}} + {y_{n - 1}}}},\quad {z_{n + 1}} = \frac{{{\alpha _{24}} + {\beta _{24 }}{e^{ - {y_n}}}}}{{{\gamma _{24}} + {z_{n - 1}}}},\] where ${\alpha _i}, {\beta _i}, {\gamma _i} $, $i = 19, 20, \cdots, 24 $ and ${x_i}, {y_i}, {z_i}, i = 0, - 1 $ are belonging to ${\mathbb{\textrm{R}}^{ \ge 0}} $. Precisely, we explore the boundedness and persistence of positive solution, existence of invariant rectangle, existence and uniqueness of positive equilibrium point, local and global dynamics about the unique positive equilibrium, and rate of convergence of these discrete-time exponential systems. Finally, theoretical results are verified numerically.

Keywords

• difference equations
• stability
• numerical simulations