Experimenting with discrete dynamical systems (Q6633402)

The authors show the power of experimental mathematics and symbolic computation to study intriguing problems on rational difference equations. This work is supplemented with the following two accompanying Maple packages:\N\N(i) \texttt{DRDS.txt}, to experiment, numerically and symbolically, with solutions of rational difference equations of any order, and for proving, if possible rigorously, but more often (if the order is three or more) semi-rigorously, global asymptotic stability;\N\N(ii) \texttt{AmalGerry.txt}, to prove rigorously (and in some cases, semi-rigorously) some conjectures about periodicity of solutions of difference equations.\N\NBoth packages are available at \url{https://sites.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/dds.html}.\N\NThe authors provide Maple commands to experiment numerically with some random rational difference equations. Further, the authors rigorously prove some fascinating conjectures by \textit{A. M. Amleh} and \textit{G. Ladas} [J. Difference Equ. Appl. 7, No. 4, 621--631 (2001; Zbl 1012.39500)] on rational difference equations. These difference equations are hard-coded in the procedure \texttt{LadadDB (x,n)}, that contains 15 interesting difference equations, some of them listed in the paper.\N\N\NConjecture 1. For any positive initial conditions \(x_{1}\), \(x_{2}\), \(x_{3}\), \N\[\Nx_{n+1}=\frac{x_{n-1}}{x_{n-1}+x_{n-2}}, \quad n\geq 3,\N\]\Nthe sequence \(\{x_{n}\}\) converges to a period-two solution of the form \(\dots, \phi, 1-\phi, \dots,\) with \(0\leq \phi\leq 1\).\N\NConjecture 2. For any positive initial conditions \(x_{1}\), \(x_{2}\), \(x_{3}\), \N\[\Nx_{n+1}=\frac{x_{n}+x_{n-2}}{x_{n-1}}, \quad n\geq 3,\N\]\Nthe sequence \(\{x_{n}\}\) converges to a period-four solution of the form \(\dots, \phi, \psi, \frac{\phi+\psi^{2}}{\phi\psi-1}, \frac{\phi^{2}+\psi}{\phi\psi-1}, \dots,\) with \(\phi, \psi \in (0, \infty)\), and \(\phi \psi >1\).\N\NConjecture 3. For any positive initial conditions \(x_{1}\), \(x_{2}\), \(x_{3}\), \N\[\Nx_{n+1}=\frac{1+x_{n-2}}{x_{n}}, \quad n\geq 3,\N\]\Nthe sequence \(\{x_{n}\}\) converges to a period-five solution of the form \(\dots, \phi, \psi, \frac{1+\phi}{\phi\psi-1}, \phi \psi -1, \frac{1+\psi}{\phi\psi-1}, \dots,\) with \(\phi, \psi \in (0, \infty)\), and \(\phi \psi >1\).\N\NConjecture 4. For any positive initial conditions \(x_{1}\), \(x_{2}\), \(x_{3}\), \N\[\Nx_{n+1}=\frac{1+x_{n}}{x_{n-1}+x_{n-2}}, \quad n\geq 3,\N\]\Nthe sequence \(\{x_{n}\}\) converges to a period-six solution of the form \(\dots, \phi, \psi, \frac{\psi}{\phi}, \frac{1}{\phi} \frac{1}{\psi}, \frac{\phi}{\psi}, \dots,\) with \(\phi, \psi \in (0, \infty)\).\N\N\NThe authors discuss some basic notions about orbits, fixed points, Jacobian matrix, global stability, etc. and mention Maple commands which are required in their experimental proofs. They also present the details of the strategy behind their proofs. For example, they present the sample outputs about rigorous and semi-rigorous proofs of global stability for certain difference equations:\N\N\begin{itemize}\N\item If you want to see 20 theorems that state that certain second-order difference equations always converge to the unique stable equilibrium (fully rigorous proofs);\N\item If you want to see 10 theorems that state that certain third-order difference equations always converge to the unique stable equilibrium (semi-rigorous proofs);\N\item If you want to see 5 theorems that state that certain fourth-order difference equations always converge to the unique stable equilibrium (semi-rigorous proofs).\N\end{itemize}\N\NMoreover, the authors try to use Maple's built-in symbolic minimizer to produce a fully rigorous proof of the minimization.\N\NThe authors also extend the work of \textit{E. Hogan} and \textit{D. Zeilberger} [J. Difference Equ. Appl. 18, No. 11, 1853--1873 (2012; Zbl 1266.39016)]\N for rigorously and semi-rigorously proving global asymptotic stability of arbitrary rational difference equations (with positive coefficients), and more generally rational transformations of the positive orthant of \(\mathbb{R}^{k}\) into itself. Finally, the authors explore automated discovery of invariants that imply that every solution is bounded.