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Linear difference equations mod 2 with applications to nonlinear difference equations. (English) Zbl 1156.39003
Consider the linear difference equation in \(\mathbb{Z}_2\) (the field of integers \(\text{mod\,}2\)) \[ x_{n+k_m}+\cdots+ x_{n+ k_1}+ x_n= \varepsilon,\tag{L} \] with integer indices, \(\varepsilon\in \{0,1\}\), natural \(m\) and \(0< k_1<\cdots< k_m\), and the nonlinear difference equation \[ z_n= F(z_{n-1},\dots, z_{n-r}),\tag{N} \] where \(F\) satisfies some nonlinear conditions. The authors extend several known results of the linear equation (L) (\(\text{mod}\,2\) with \(T\)-periodic solutions) to the nonlinear equation (N) and compile them for applications to the semicycle analysis of the nonlinear difference equation (N). For the calculation of \(T\), four methods are presented. A further application concerns rational functions in the field of integers \(\text{mod\,}2\).

MSC:
39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
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