Basins of attraction of equilibrium points of monotone difference equations. (English) Zbl 1189.39024

The article deals with the difference equation
\[ x_{n+1} = f(x_n,x_{n-1}), \qquad n = 0,1,\dots,\tag{*} \]
with \(f \in C[I \times I,I]\) (\(I\) is an interval in \({\mathbb R}\)); it is assumed that \(f\) is increasing in both variables. More precisely, the equilibria and their basins of attraction are studied. The main results are the following: 7mm
If (*) has no prime period-two solutions, then every bounded solution of (*) converges to an equilibrium.
If there exist two equilibrium points \(\overline{x}_1, \overline{x}_2\), \(0 \leq \overline{x}_1 < \overline{x}_2\), then the box \((\overline{x}_1,\overline{x}_2)^2\) is a part of the basin of attraction of an equilibrium, \(\overline{x}_1\) if \((x - \overline{x}_1)(f(x,x) - x)< 0\) for \(x \in (\overline{x}_1,\overline{x}_2)\) is satisfied and \(\overline{x}_2\) if \((x - \overline{x}_2)(f(x,x) - x) < 0\) for \(x \in (\overline{x}_1,\overline{x}_2)\) is satisfied.
If (*) has no minimal period-two solutions and \(E_1(x_1,y_1)\), \(E_2(x_2,y_2)\), \(E_3(x_3,y_3)\) are three consecutive equilibrium points, \((x_1,y_1) \preceq (x_2,y_2) \preceq(x_3,y_3)\), \(E_1\), \(E_3\) are saddle points, \(E_2\) is a local attractor, then the basin of attraction \({\mathcal B}(E_2)\) is the region between the global stable manifolds \({\mathcal W}^s(E_1)\) and \({\mathcal W}^s(E_3)\) and the basins of attraction \({\mathcal B}(E_2) = {\mathcal W}^s(E_1)\) and \({\mathcal B}(E_3) = {\mathcal W}^s(E_3)\) are exactly the global stable manifolds of \(E_1\) and \(E_2\). [The latter statement is vague and puzzles: the equilibrium of (*) are real numbers of \(I\); \(E_1\), \(E_2\), \(E_3\) are not elements of \(I\).]
The following examples with the corresponding nice illustrations are considered:
\[ \begin{aligned} x_{n+1} &= \tfrac12(x_n + x_{n-1} + \sin x_{n-1}),\\ x_{n+1} &= \tfrac12(x_n + x_{n-1} + \sin x_n + \sin x_{n-1}),\\ x_{n+1} &= x_n^3 + x_{n-1}^3;\end{aligned} \]
and some other examples are also presented. Also, some analogues of the main results about equation (*) are formulated for the equation
\[ x_{n+1} = f(x_n,x_{n-1},\dots,x_{n-k+1}), \qquad n = 0,1,\dots \]


39A30 Stability theory for difference equations
39A10 Additive difference equations
39A23 Periodic solutions of difference equations