Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Asymptotics of the Arnold tongues in problems at infinity

Pages: 989 - 1011, Volume 20, Issue 4, April 2008      doi:10.3934/dcds.2008.20.989

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Victor Kozyakin - Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane 19, Moscow 127994 GSP-4, Russian Federation (email)
Alexander M. Krasnosel’skii - Institute for Information Transmission Problems, Russian Academy of Sciences, 19 Bol.Karetny Lane, Moscow GSP-4, 127994, Russia; National Research University Higher School of Economics, 20 Myasnitskaya Street, Moscow 101000, Russian Federation (email)
Dmitrii Rachinskii - Department of Applied Mathematics, University College, Cork, Ireland (email)

Abstract: 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:  Bifurcation at infinity, periodic trajectory, Arnold tongue, subfurcation, positively homogeneous nonlinearity, saturation, discrete time system, Poincare map, invariant set, subharmonics, rotation of vector fields.
Mathematics Subject Classification:  37G15, 34C25, 34B15.

Received: February 2007;      Revised: October 2007;      Available Online: January 2008.