Asymptotics of the Arnold tongues in problems at infinity (Q946968)

The authors study bifurcations of periodic trajectories from infinity for the discrete time system \(x_{k+1}=U(x_k;\lambda),x\in \mathbb{R}^n, N\geq 2, \lambda\in D\subset C\), supposing that for \(x\) with sufficiently large norm the mapping \(U\) is continuous with its arguments and has the form \[ U(x,\lambda)=A(\lambda)x+\Phi(x;\lambda)+\xi(x;\lambda). \] Here \(A(\lambda)x\) is the linear part, \(\Phi\) is a bounded positively homogeneous nonlinearity and \(\xi\) is a small term. The aim of the article is to study the set \(\mathcal{P}_{\lambda}\subset \mathbb{R}^N\) of all periodic points of the system, in particular Arnol'd tongues: the sets of parameter values for which the large-amplitude periodic trajectories exist.