Arnold tongues in the problem of large-amplitude periodic trajectories
Consider a finite-dimensional discrete time dynamical system \(x_{k+1}=U_{\lambda}x_k\), \(x\in {\mathbb R}^n\) depending on a complex parameter \(\lambda\). In a neighbourhood of an equilibrium (say, at \(x_*=0\)) its properties are determined by its linear part \(A_{\lambda}x\). If the matrix \(A_{\lambda}\) has eigenvalues near the unit circle, then in a neighbourhood of \(0\) the system may have periodic orbits with different periods. Periodic regimes occur for eigenvalues of the matrix \(A_{\lambda}\) from certain domains \(U_q\) (with vertices at the points \(e^{2\pi qi}\)) called Arnold tongues, where \(q\in {\mathbb Q}\). The paper studies such tongues at infinity, assuming that the operator \(U_{\lambda}\) is of the form \(A_{\lambda}x+\)bounded nonlinearity.
finite-dimensional discrete time dynamical system
scientific article; zbMATH DE number 5824751