Generations of solvable discrete-time dynamical systems

TL;DR

This paper introduces a novel technique to generate new solvable discrete-time dynamical systems from existing ones, enabling the creation of complex models with arbitrary constants and special time evolution properties.

Contribution

The paper presents a new algebraic method to produce generations of solvable discrete-time systems, expanding the class of models with known solutions and diverse behaviors.

Findings

Generated new classes of solvable systems with arbitrary constants

Identified special cases with isochronous and asymptotically isochronous behavior

Demonstrated the technique with illustrative examples

Abstract

A technique is introduced which allows to generate -- starting from any solvable discrete-time dynamical system involving N time-dependent variables -- new, generally nonlinear, generations of discrete-time dynamical systems, also involving N time-dependent variables and being as well solvable by algebraic operations (essentially by finding the N zeros of explicitly known polynomials of degree N). The dynamical systems constructed using this technique may also feature large numbers of arbitrary constants, and they need not be autonomous. The solvable character of these models allows to identify special cases with remarkable time evolutions: for instance, isochronous or asymptotically isochronous discrete-time dynamical systems. The technique is illustrated by a few examples.