17. 1. 2024.

1

This paper investigates an autonomous predator-prey system of difference equations with three equilibrium points and exhibits chaos in the sense of Li-Yorke in the positive equilibrium point. Numerical simulations are presented to illustrate our results.

1. 11. 2021.

0

This paper investigates the local and global character of the unique positive equilibrium of a mixed monotone fractional second-order difference equation with quadratic terms. The corresponding associated map of the equation decreases in the first variable, and it can be either decreasing or increasing in the second variable depending on the corresponding parametric values. We use the theory of monotone maps to study global dynamics. For local stability, we use the center manifold theory in the case of the non-hyperbolic equilibrium point. We show that the observed equation exhibits three types of global behavior characterized by the existence of the unique positive equilibrium, which can be locally stable, non-hyperbolic when there also exist infinitely many non-hyperbolic and stable minimal period-two solutions, and a saddle. Numerical simulations are carried out to better illustrate the results.

12. 2. 2020.

4

We investigate global dynamics of the equation\begin{equation*}x_{n+1}=\frac{x_{n-1}+F}{ax_{n}^2+f},\text{ \ }n=0,1,2,...,\end{equation*}where the parameters $a,F$ and $f$ are positive numbers and the initial conditions $x_{-1},x_{0}$ are arbitrary nonnegative numbers such that $x_{-1}+x_{0}>0$. The existence and local stability of the unique positive equilibrium are analyzed algebraically. We characterize the global dynamics of this equation with the basins of attraction of its equilibrium point and periodic solutions.

2. 12. 2019.

2

We investigate the local and global character of the unique equilibrium point of certain homogeneous fractional difference equation with quadratic terms. The existence of the period-two solution in one special case is given. Also, in this case the local and global stability of the minimal period-two solution for some special values of the parameters are given. AMS Subject Classifications: 39A10, 39A20, 39A23, 39A30.

15. 1. 2017.

3

By using the Kolmogorov–Arnold–Moser theory, we investigate the stability of the equilibrium solution of the difference equation un+1=A+Bun+un2(1+Dun)un−1,n=0,1,2,… where A,B,D > 0,u−1,u0>0. We also use the symmetries to find effectively the periodic solutions with feasible periods. Copyright © 2016 John Wiley & Sons, Ltd.

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