Written: May/June 2023

Maple package

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

Sample Input and Output for DRDS.txt

• If you want to see second-order rational recurrences with periods up to the sixth (including a rediscovery of an infinite family that includes the famous Lynnes recurrenve x[n+1]=(1+x[n])/x[n-1]. It can be easily seen that by a change of (dependent) variable, the deceptively general form
  (a[1]*b[2]*x[n]-b[0]*b[2]*x[n-1]+a[1]^2+a[1]*b[0]-b[0]^2)/b[2]/(b[2]*x[n-1]+b[0])
  can be brought to the Lynnes form. Similary for the other ones in this file.
  The input file yields the output file.

• If you to see 10 computer-generated proofs of global attractors for numerous (randomly chosen) one-dimensional rational transformation (with quadratic numerators and denominators, all with positive coefficients so that [0,infinity] goes to [0,infinity]
  The input file yields the output file.

• If you 30 computer-generated proofs of global attractors for numerous (randomly chosen) two-dimensional rational transformation (with degree one of both numerators and denominators) all with positive coefficients so that [0,infinity]² goes to [0,infinity]² goes into itself
  The input file yields the output file.

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  • If you 30 computer-generated CONJECTURED global attractors for numerous (randomly chosen) one-dimensional rational transformation
    The input file yields the output file.

  • If you 30 computer-generated CONJECTURED global attractors for numerous (randomly chosen) two-dimensional rational transformation
    The input file yields the output file.

  • If you 30 computer-generated CONJECTURED global attractors for numerous (randomly chosen) three-dimensional rational transformation
    The input file yields the output file.

  • If you 30 computer-generated CONJECTURED global attractors for numerous (randomly chosen) four-dimensional rational transformation
    The input file yields the output file.

  • If you 30 computer-generated CONJECTURED global attractors for numerous (randomly chosen) fivedimensional rational transformation
    The input file yields the output file.

  ---

• If you see many PERIODIC RECURRENCES (Like Lynnes')
  The input file yields the output file.

• If you 20 computer-generated rigorously (and sometimes semi-rigorously) proved results that certain randomly-chosen SECOND-ORDER rational recurrences (with numerator and deminators with positive coefficients) always converge to the same limit The input file yields the output file.

• If you 20 computer-generated rigorously (and sometimes semi-rigorously) proved results that certain randomly-chosen THIRD-ORDER rational recurrences (with numerator and deminators with positive coefficients) always converge to the same limit The input file yields the output file.

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