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Global behaviour of a second-order nonlinear difference equation. (English) Zbl 1232.39014

The article deals with the Cauchy problem

x n+1 =x n-1 a+bx n x n-1 ,(x -1 ,x 0 ) 2 ,n=0,1,2,,

where a,b are reals. It is proved that the general solution to this equation (beside some exceptional cases) is presented in the form

x 2k+2 =x 0 i=0 k h(2i+1),x 2k+1 =x -1 i=0 k h(2i),


h(n)=a n (1-a)+α(-a n ) a n+1 (1-a)+α(1-a n+1 ,ifa1,1+αn 1+α(n+1),ifa=1·

(α=bx -1 x 0 ). Further, a complete analysis of the asymptotic behavior of these solutions in different cases (a=-1; |a|1,a-1; |a|<1) is given, the behavior of these solutions near bifurcation points (a=-1 and a=1) s described and the stability properties of nonzero periodic solutions is investigated.

39A20Generalized difference equations
39A30Stability theory (difference equations)
39A28Bifurcation theory (difference equations)
39A22Growth, boundedness, comparison of solutions (difference equations)