On the accumulation points of non-periodic orbits of a difference equation of fourth order

Abstract

In this paper, we are interested in analyzing the dynamics of the fourth-order difference equation $x_{n+4} = \max\{x_{n+3},x_{n+2},x_{n+1},0\}-x_n$, with arbitrary real initial conditions. We fully determine the accumulation point sets of the non-periodic solutions that, in fact, are configured as proper compact intervals of the real line. This study complements the previous knowledge of the dynamics of the difference equation already achieved in [M. Cs\"ornyei, M. Laczkovich, Monatsh. Math. 132 (2001), 215-236] and [A. Linero Bas, D. Nieves Rold\'an, J. Difference Equ. Appl. 27 (2021), no. 11, 1608-1645].