Kozyakin, Victor S.; Krasnosel’skii, Alexander M.; Rachinskii, Dmitrii I. Arnold tongues for bifurcation from infinity. (English) Zbl 1152.39006 Discrete Contin. Dyn. Syst., Ser. S 1, No. 1, 107-116 (2008). The authors study bifurcations of large periodic trajectories of the system \(x_{k+1}=U( x_{k};\lambda) ,\) \(x\in\mathbb{R}^{N},\) \(N\geq2,\) where \(\lambda\) is a complex parameter. For sufficiently large \(| x| ,\) the map \(U\) is supposed continuous with respect to the set of its arguments and \[ U( x;\lambda) =A( \lambda) x+\Phi( x;\lambda) +\xi( x;\lambda), \]where \(A( \lambda) x\) is the principal linear part, \(\Phi( \cdot;\lambda) \) is a bounded positively homogeneous nonlinearity of order \(0,\) and \(\xi( \cdot;\lambda) \) is a small part (it tends to zero at infinity).The authors find the sets of parameter values for which the large-amplitude \(n\)-periodic trajectories exist for a fixed \(n.\) They show that in the related problems on small periodic orbits near zero, Arnold tongues are more narrow. Reviewer: N. C. Apreutesei (Iaşi) MSC: 39A11 Stability of difference equations (MSC2000) 37G10 Bifurcations of singular points in dynamical systems 39A12 Discrete version of topics in analysis 37C27 Periodic orbits of vector fields and flows Keywords:periodic trajectory; bifurcation at infinity; Arnold tongue; positively homogeneous nonlinearity; saturation; Poincaré map PDFBibTeX XMLCite \textit{V. S. Kozyakin} et al., Discrete Contin. Dyn. Syst., Ser. S 1, No. 1, 107--116 (2008; Zbl 1152.39006) Full Text: DOI