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

Antonio Linero Bas, Víctor Mañosa, Daniel Nieves Roldán

TL;DR

This paper resolves the accumulation behavior of non-periodic orbits for the max-type, fourth-order difference equation $x_{n+4}=\max\{x_{n+3},x_{n+2},x_{n+1},0\}-x_n$, showing that every non-periodic trajectory is dense in a compact interval. By normalizing to $x=\max_n x_n$ and enforcing $x\ge w\ge y\ge z\ge 0$ with $\frac{w-z}{x}$ irrational, the accumulation set is explicitly $[\min\{w-x,-z\},x]$. The authors develop a route-based description of the dynamics, identify key linear forms of the type $t x+y-s(w-z)$ that arise along the routes, and apply Kronecker-type density results to prove both non-negative and non-positive terms are dense in their respective subintervals. The results complement prior periodicity and boundedness analyses and reveal a rich Diophantine structure behind the non-periodic dynamics, including a newly identified invariant $V_2$ and evidence for a potential third independent first integral.

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. Csörnyei, M. Laczkovich, Monatsh. Math. 132 (2001), 215-236] and [A. Linero Bas, D. Nieves Roldán, J. Difference Equ. Appl. 27 (2021), no. 11, 1608-1645].

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

TL;DR

This paper resolves the accumulation behavior of non-periodic orbits for the max-type, fourth-order difference equation , showing that every non-periodic trajectory is dense in a compact interval. By normalizing to and enforcing with irrational, the accumulation set is explicitly . The authors develop a route-based description of the dynamics, identify key linear forms of the type that arise along the routes, and apply Kronecker-type density results to prove both non-negative and non-positive terms are dense in their respective subintervals. The results complement prior periodicity and boundedness analyses and reveal a rich Diophantine structure behind the non-periodic dynamics, including a newly identified invariant and evidence for a potential third independent first integral.

Abstract

In this paper, we are interested in analyzing the dynamics of the fourth-order difference equation , 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. Csörnyei, M. Laczkovich, Monatsh. Math. 132 (2001), 215-236] and [A. Linero Bas, D. Nieves Roldán, J. Difference Equ. Appl. 27 (2021), no. 11, 1608-1645].
Paper Structure (13 sections, 9 theorems, 96 equations, 3 figures, 2 tables)

This paper contains 13 sections, 9 theorems, 96 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Evolution of non-negative tuples starting from Case $C4$
  4. First terms of an orbit through the routes
  5. Evolution of a general tuple through the routes
  6. Proof of Theorem \ref{['t:teoprincipal']}
  7. Density of the non-negative terms
  8. Density of the non-positive terms
  9. Evolution of a non-periodic orbit through the cases $C_i$
  10. Density of the non-positive terms in $[w-x,0]$
  11. Density of the non-positive terms in $[-z,0]$
  12. Proof of Theorem \ref{['t:teoprincipal']}
  13. Conclusions and other future directions

Key Result

Theorem A

Given real initial conditions $(x_1,x_2,x_3,x_4)$ that generate a non-periodic orbit under Equation (Eq_main), its set of accumulation points is a compact interval. Even more, the tuple $(x_1,x_2,x_3,x_4)$ is equivalent to some tuple of initial conditions $(x,y,z,w),$ with $x=\max\{x_n:n\geq 1\}$, $

Figures (3)

  • Figure 1: Evolution of the cases under the iteration of Equation (\ref{['Eq_main']}).
  • Figure 2: Two views of the orbit of the map $F$ with initial conditions $(2\sqrt{2},2,0,1)$
  • Figure 3: Two views of the orbit of the map $F$ with initial conditions $(10\sqrt{3}+\sqrt{2},1,2,0)$

Theorems & Definitions (17)

  • Definition 1
  • Theorem A
  • Claim 1
  • Proposition 1
  • Proposition 2
  • proof
  • Theorem 2
  • Corollary 1
  • Remark 1
  • Remark 2
  • ...and 7 more