Experimenting with Discrete Dynamical Systems

By George Spahn and Doron Zeilberger


.pdf    .tex   

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)


Dedicated to Saber Elaydi on his 80th birthday, and to Gerry Ladas on his 85th birthday


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


Maple packages


Sample Input and Output for DRDS.txt


Sample Input and Output for AmalGerry.txt

  • If you want to see a semi-rigorous proof to Conjecture 1 (due to Amal Amleh and Gerry Ladas) mentioned in the paper


    the input file yields the output file.

  • If you want to see a semi-rigorous proof to Conjecture 2 (due to Amal Amleh and Gerry Ladas) mentioned in the paper


    the input file yields the output file.

  • If you want to see a semi-rigorous proof to Conjecture 3 (due to Amal Amleh and Gerry Ladas) mentioned in the paper


    the input file yields the output file.

  • If you want to see a semi-rigorous proof to Conjecture 4 (due to Amal Amleh and Gerry Ladas) mentioned in the paper


    the input file yields the output file.

  • If you want to see a FULLY rigorous proof to Conjecture 1 (due to Amal Amleh and Gerry Ladas) mentioned in the paper


    the input file yields the output file.


    Sample Input and Output for LOZITSKY.txt

    • If you want to see a the periodic Lynnes-style recurrences produced by Evgeni Lozitsky in his paper, plus one more of order 2 and period 18
      the input file yields the input file file


    A Challenge to Manuel Kauers

    In the paper we advocated using semi-rigorous proofs for rational recurrences of order 3 and up, since the proof boils down to proving that a certain (complicated) polynomial is non-negative in Rk. But with human ingenuity, and some tricks-of-the-trade, humans can do it (of course assisted by computers).

    One of us (DZ) offered to donate $100 dollars to the OEIS in honor of Manuel Kauers, if he would meet this challenge, that he brillinatlly met. A donation to the OEIS, in Manuel' honor has been made, as promised.


    Doron Zeilberger's Home Page