Periodicity and convergence for \(x_{n+1}=| x_{n}-x_{n-1}|\).
The author proves that each solution \(\{x_n\}\) of the equation \(x_{n+1}=| x_{n}-x_{n-1}| \) is either eventually periodic with period 3 or converges to zero. The case depends on whether the ratio \(x_0/x_{-1}\) of the initial values is rational or irrational. The sequence of ratios \(r_n=x_n/x_{n-1}\), \(n=0,1,\ldots\), satisfies the difference equation \(r_{n+1}=\phi(r_n)=| 1/r_n-1| \) that has \(p\)-cycles for all integer \(p\neq 3\). Each cycle is explicitly determined by Fibonacci numbers. A unique positive fixed point of \(\phi\) is shown to be a snap-back repeller. According to the result by \textit{F. R. Marotto} [J. Math. Anal. Appl. 63, 199--223 (1978; Zbl 0381.58004)] it means that the irrational ratios \(r_n\) behave chaotically.
scientific article; zbMATH DE number 2056493