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

The article deals with the difference equation

${x}_{n+1}=f\left({x}_{n},{x}_{n-1}\right),\phantom{\rule{2.em}{0ex}}n=0,1,\cdots ,\phantom{\rule{2.em}{0ex}}\left(*\right)$

with $f\in C\left[I×I,I\right]$ ($I$ is an interval in $ℝ$); 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:

If (*) has no prime period-two solutions, then every bounded solution of (*) converges to an equilibrium.

If there exist two equilibrium points ${\overline{x}}_{1},{\overline{x}}_{2}$, $0\le {\overline{x}}_{1}<{\overline{x}}_{2}$, then the box ${\left({\overline{x}}_{1},{\overline{x}}_{2}\right)}^{2}$ is a part of the basin of attraction of an equilibrium, ${\overline{x}}_{1}$ if $\left(x-{\overline{x}}_{1}\right)\left(f\left(x,x\right)-x\right)<0$ for $x\in \left({\overline{x}}_{1},{\overline{x}}_{2}\right)$ is satisfied and ${\overline{x}}_{2}$ if $\left(x-{\overline{x}}_{2}\right)\left(f\left(x,x\right)-x\right)<0$ for $x\in \left({\overline{x}}_{1},{\overline{x}}_{2}\right)$ is satisfied.

If (*) has no minimal period-two solutions and ${E}_{1}\left({x}_{1},{y}_{1}\right)$, ${E}_{2}\left({x}_{2},{y}_{2}\right)$, ${E}_{3}\left({x}_{3},{y}_{3}\right)$ are three consecutive equilibrium points, $\left({x}_{1},{y}_{1}\right)⪯\left({x}_{2},{y}_{2}\right)⪯\left({x}_{3},{y}_{3}\right)$, ${E}_{1}$, ${E}_{3}$ are saddle points, ${E}_{2}$ is a local attractor, then the basin of attraction $ℬ\left({E}_{2}\right)$ is the region between the global stable manifolds ${𝒲}^{s}\left({E}_{1}\right)$ and ${𝒲}^{s}\left({E}_{3}\right)$ and the basins of attraction $ℬ\left({E}_{2}\right)={𝒲}^{s}\left({E}_{1}\right)$ and $ℬ\left({E}_{3}\right)={𝒲}^{s}\left({E}_{3}\right)$ 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:

$\begin{array}{cc}\hfill {x}_{n+1}& =\frac{1}{2}\left({x}_{n}+{x}_{n-1}+sin{x}_{n-1}\right),\hfill \\ \hfill {x}_{n+1}& =\frac{1}{2}\left({x}_{n}+{x}_{n-1}+sin{x}_{n}+sin{x}_{n-1}\right),\hfill \\ \hfill {x}_{n+1}& ={x}_{n}^{3}+{x}_{n-1}^{3};\hfill \end{array}$

and some other examples are also presented. Also, some analogues of the main results about equation (*) are formulated for the equation

${x}_{n+1}=f\left({x}_{n},{x}_{n-1},\cdots ,{x}_{n-k+1}\right),\phantom{\rule{2.em}{0ex}}n=0,1,\cdots$

##### MSC:
 39A30 Stability theory (difference equations) 39A10 Additive difference equations 39A23 Periodic solutions (difference equations)