DCDS-S
Arnold tongues for bifurcation from infinity
Victor S. Kozyakin Alexander M. Krasnosel’skii Dmitrii I. Rachinskii
We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$, $x\in\R^{N}$, with a complex parameter $\lambda$. The map $U(\cdot;\lambda)$ at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller part. We describe the sets of parameter values for which the large-amplitude $n$-periodic trajectories exist for a fixed $n$. In the related problems on small periodic orbits near zero, similarly defined parameter sets, known as Arnold tongues, are more narrow.
keywords: Arnold tongue positively homogeneous nonlinearity discrete time system bifurcation at infinity Poincare map. Periodic trajectory saturation
DCDS
Asymptotics of the Arnold tongues in problems at infinity
Victor Kozyakin Alexander M. Krasnosel’skii Dmitrii Rachinskii
We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$, $x\in\R^{N}$, with a complex parameter $\lambda$, and study their trajectories of large amplitudes. The expansion of the map $U(\cdot;\lambda)$ at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller vanishing part. We study Arnold tongues: the sets of parameter values for which the large-amplitude periodic trajectories exist. The Arnold tongues in problems at infinity generically are thick triangles [4]; here we obtain asymptotic estimates for the length of the Arnold tongues and for the length of their triangular part. These estimates allow us to study subfurcation at infinity. In the related problems on small-amplitude periodic orbits near an equilibrium, similarly defined Arnold tongues have the form of narrow beaks. For standard pictures associated with the Neimark-Sacker bifurcation of smooth discrete time systems at an equilibrium, the Arnold tongues have asymptotically zero width except for the strong resonance points. The different shape of the tongues in the problem at infinity is due to the non-polynomial form of the principal homogeneous nonlinear term of the map $U(\cdot;\lambda)$: this form implies non-degeneracy of the nonlinear terms in the expansion of the map iterations and non-degeneracy of the corresponding resonance functions.
keywords: periodic trajectory discrete time system Bifurcation at infinity Poincare map Arnold tongue subfurcation positively homogeneous nonlinearity saturation invariant set rotation of vector fields. subharmonics

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