The dynamical behavior of a system of max-type difference equations model

Abstract

Difference equation as a discrete model has been widely used in the fields of algorithm analysis,control theory, computer science, biology, economics and so on. In this paper, we mainly study the dynamical behavior of the system of high order max-type difference equations model,

$$x_n=\max \{\frac{A}{y_{n-1}},x_{n-k}\},y_n=\max \{\frac{B}{x_{n-1}},y_{n-k}\},n=1,2,\mathrm{\dots },$$

where $A$ and $B$ are non-negative real numbers, initial values are positive real numbers. We use mathematical induction to study the periodic properties of the solutions when $AB=0$ and $AB\ne 0$ . Our conclusion is that: (1) Every positive solution of the difference equations model is periodic with period $k$ if $AB=0$ ; (2) Every positive solution of the difference equations model is eventually periodic with period $k$ if $AB\ne 0$ . At last we give some numberical examples about the system to verify our theoretical results. The conclusions of this paper complement and generalize the existing results.