Stability of the Gumowski-Mira equation with period-two coefficient
The authors investigate the stability of solutions to the nonlinear second-order difference equation \[ (\ast)\qquad y_{n+1} = \frac{y_n}{b_n+y_n^2}-y_{n-1}, \] known as Gumowski-Mira equation. Here, the sequence \((b_n)\) is assumed to be \(2\)-periodic with nonnegative values \(\alpha,\beta\). A linear stability analysis is performed for the equilibrium points of \((\ast)\), and using invariants their stability is established for \(\alpha\beta\not\in\{0,\frac{1}{4}\}\). The remaining part of the paper deals with the case \(\alpha=0\), where the forbidden set of \((\ast)\) is determined, i.e., the set of initial values \(y_{-1},y_0\) such that \(b_n+y_n^2=0\) for one \(n\geq-1\). It is shown that the subsequence of odd indices \((y_{2n-1})\) is unbounded, and depending on the sign of \(y_{-1}y_0-\beta\), the even subsequence \((y_{2n})\) converges to \(0\) or to a period-two sequence. Finally, a result concerning the case \(\alpha=\beta=0\) is quoted from the literature.
nonlinear second-order difference equation
scientific article; zbMATH DE number 2174939