zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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,

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