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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(x n ,x n-1 ),n=0,1,,(*)

with fC[I×I,I] (I is an interval in ); 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 x ¯ 1 ,x ¯ 2 , 0x ¯ 1 <x ¯ 2 , then the box (x ¯ 1 ,x ¯ 2 ) 2 is a part of the basin of attraction of an equilibrium, x ¯ 1 if (x-x ¯ 1 )(f(x,x)-x)<0 for x(x ¯ 1 ,x ¯ 2 ) is satisfied and x ¯ 2 if (x-x ¯ 2 )(f(x,x)-x)<0 for x(x ¯ 1 ,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 )(x 2 ,y 2 )(x 3 ,y 3 ), E 1 , E 3 are saddle points, E 2 is a local attractor, then the basin of attraction (E 2 ) is the region between the global stable manifolds 𝒲 s (E 1 ) and 𝒲 s (E 3 ) and the basins of attraction (E 2 )=𝒲 s (E 1 ) and (E 3 )=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:

x n+1 =1 2(x n +x n-1 +sinx n-1 ),x n+1 =1 2(x n +x n-1 +sinx n +sinx n-1 ),x n+1 =x n 3 +x n-1 3 ;

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 ,,x n-k+1 ),n=0,1,

MSC:
39A30Stability theory (difference equations)
39A10Additive difference equations
39A23Periodic solutions (difference equations)