Periodic solutions of second-order nonlinear difference equations containing a small parameter. III: Perturbation theory
The paper is the continuation of two previous ones, concerning the same subject and by the same author [ibid. 316, 273-277 (1983; Zbl 0529.39002) and ibid. 320, 169-174 (1985; Zbl 0589.39004)]. Here the purpose is to construct first and second approximations to the solutions of the following class of nonlinear difference equations \[ x_{k+1}-(2 \cos \phi)x_ k+x_{k-1}=\epsilon f, \] where \(\phi\) is a given constant; f is a polynomial function of \((x_{k+1},x_ k,x_{k-1})\); \(\epsilon\) is a small positive parameter \((0<\epsilon \ll 1)\) and \(\{\) \(k\}\) is the set of positive integers. The techniques exploited are the perturbation techniques of the Lindstedt-Poincaré method.
scientific article; zbMATH DE number 3951177