Written: June 21, 2023 ; Last update of this web-page: Nov. 24, 2023

Appeared in J. Difference Equations and Applications [Special Elaydi issue] (on-line before print, volume and page tbd)

We demonstrate the power of Experimental Mathematics and Symbolic Computation to study intriguing problems on rational difference equations, studied extensively by Difference Equations giants, Saber Elaydi and Gerry Ladas (and their students and collaborators). In particular we rigorously prove some fascinating conjectures made by Amal Amleh and Gerry Ladas back in 2000. For other conjectures we are content with semi-rigorous proofs. We also extend the work of Emilie Purvine (formerly Hogan) and Doron Zeilberger for rigorously and semi-rigorously proving global asymptotic stability of arbitrary rational difference equations (with positive coefficients).

Added Sept. 12, 2023: Read the interesting message from Evgeni Lozitsky

Added Sept. 19, 2023: Read the equally interesting Second message from Evgeni Lozitsky

Added Oct. 4, 2023: Evgeni Lozitsky wrote it up in a very interesting paper

Added Nov. 23, 2023: Inspired by Evgeni Lozitsky, who used Mathematica, we reproduced and extended his results by writing a Maple package LOZITSKY.txt (see below)

Added Dec. 6, 2023: Evgeni Lozitsky found recurrences of order thirty!
See third message from Evgeni Lozitsky

Added March 20, 2024: Evgeni Lozitsky found many other intriguing conjectures!
See fourth message from Evgeni Lozitsky

Maple packages

• DRDS.txt, a Maple package to investigate Discrete Rational Dynamical Systems

• AmalGerry.txt, a Maple package specifically targeted to prove, sometimes rigorously, but also semi-rigorously, the intriguing conjectures by Amal Amleh and Gerry Ladas discussed in the paper.

• Postpublication Maple package (posted Nov. 24, 2023)

  LOZITSKY.txt,

Sample Input and Output for DRDS.txt

• If you want to see 20 FULLY-RIGOROUSLY-PROVED theorems about global convergence of certain SECOND-ORDER rational recurrences then
  the input file yields the output file.

• If you want to see 10 SEMI-RIGOROUSLY-PROVED theorems about global convergence of certain THIRD-ORDER rational recurrences then
  the input file yields the output file.

• If you want to see 5 SEMI-RIGOROUSLY-PROVED theorems about global convergence of certain FOURTH-ORDER rational recurrences then
  the input file yields the output file.
  WARNING: BIG FILE!

• If you want to see a rigorous proof that the orbit of the generalized Lynnes equation

  x[n+1]=(p+x[n])/x[n-1], where p is positive, and positive initial conditions is ALWAYS bounded, and not only that the quarting equation whose middle roots give you the min and max of the orbit

  
  the input file yields the output file.

• If you want to see a rigorous proof that the orbit of the generalized LADAS equation

  x[n+1]=(p+x[n]+x[n-1])/x[n-2], where p is positive, and positive initial conditions is ALWAYS bounded, and not only that the surface in 3D space whose projection to the X-axis gives you the min and max of the orbit

  
  the input file yields the output file.

• If you want to see invariants of the k-th order recurrence equations

  x[n+1]=(p+x[n]+x[n-1]+...+x[n-k+2])/x[n-k+1],

  where p is positive, for k from 1 to 8

  
  the input file yields the output file.

Sample Input and Output for AmalGerry.txt