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Stability of a class of nonlinear difference equations. (English) Zbl 0923.39005

Using KAM theory, the authors investigate the stability nature of the zero equilibrium of the system of two nonlinear difference equations \[ \left.\begin{matrix} x_{n+1}=a_1x_n+b_1y_n+f(c_1x_n+c_2y_n)=F_1(x_n,y_n)\\ y_{n+1}=a_2x_n+b_2y_n+f(c_1x_n+c_2y_n)=F_2(x_n,y_n)\end{matrix} \right\}, \quad n=0,1,\dots, \tag{*} \] \(n=0,1,2,\dots\), where \(a_i,b_i,c_i\), \(i=1,2\), are real constants and \(f:\mathbb{R}\to\mathbb{R}\) is a \(C^\infty\) function. It is supposed that the map \(F=(F_1,F_2)\) satisfies the following condition \((H)\): \(F\) is an area preserving map with an elliptic fixed point at the origin and eigenvalues \(\lambda,\overline\lambda\) such that \(\lambda^k\neq 1\), \(k=3,4\). By Proposition 1 it is shown that \(F\) satisfies condition \((H)\) if and only if \(a_1b_2-a_2b_1=1\), \(| a_1+b_2|<2\), \(a_1+b_2\neq -1\), \((a_1-a_2)c_2=(b_1-b_2)c_1\). In Proposition 2 conditions on \(a_i,b_i,c_i\), \(i=1,2\), and the function \(f\) are given so that the zero equilibrium of (*) is stable.

MSC:

39A11 Stability of difference equations (MSC2000)
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