March  2008, 1(1): 107-116. doi: 10.3934/dcdss.2008.1.107

Arnold tongues for bifurcation from infinity

1. 

Institute for Information Transmission Problems, Russian Academy of Sciences, 19 Bol.Karetny Lane, Moscow GSP-4, 127994, Russian Federation

2. 

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

3. 

Department of Applied Mathematics, University College Cork, Cork, Ireland

Received  September 2006 Revised  January 2007 Published  December 2007

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.
Citation: Victor S. Kozyakin, Alexander M. Krasnosel’skii, Dmitrii I. Rachinskii. Arnold tongues for bifurcation from infinity. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 107-116. doi: 10.3934/dcdss.2008.1.107
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