A note on an open conjecture in rational dynamical systems

TL;DR

This paper disproves a long-standing conjecture in rational dynamical systems by constructing counterexamples, using subenergy functions and properties of Todd's difference equation, and introduces new Chebyshev approximation results.

Contribution

The paper provides the first negative answer to the open conjecture on boundedness in rational dynamical systems, employing novel analytical methods.

Findings

Disproved the conjecture that bounded solutions occur iff eta=lambda.

Constructed counterexamples using subenergy functions.

Presented new Chebyshev approximation results.

Abstract

Recently ,mathematicians have been interested in studying the theory of discrete dynamical system, specifically difference equation, such that considerable works about discussing the behavior properties of its solutions (boundedness and unboundedness) are discussed and published in many areas of mathematics which involves several interesting results and applications in applied mathematics and physics ,One of the most important discrete dynamics which is became of interest for researchers in the field is the rational dynamical system .In this paper we give a negative answer to the eight open conjecture in rational dynamical system proposed by G.Ladas and Palladino many years ago which states : Assume $\alpha,\beta, \lambda \in [0,\infty)$. Then every positive solution of the difference equation \\: \begin{align*} z_{n+1}=\frac{\alpha+z_{n}\beta +z_{n-1}\lambda}{z_{n-2}},\quad n=0,1,\ldots \end{align*} is bounded if and only if $\beta=\lambda$. We will use a construction of subenergy function and some properties of Todd's difference equation to disprove that conjecture in general.Some new results (Chebychev approximation) and analysis regarding that open conjecture are presented.