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Arnold tongues for bifurcation from infinity. (English) Zbl 1152.39006

The authors study bifurcations of large periodic trajectories of the system \(x_{k+1}=U( x_{k};\lambda) ,\) \(x\in\mathbb{R}^{N},\) \(N\geq2,\) where \(\lambda\) is a complex parameter. For sufficiently large \(| x| ,\) the map \(U\) is supposed continuous with respect to the set of its arguments and
\[ U( x;\lambda) =A( \lambda) x+\Phi( x;\lambda) +\xi( x;\lambda), \]
where \(A( \lambda) x\) is the principal linear part, \(\Phi( \cdot;\lambda) \) is a bounded positively homogeneous nonlinearity of order \(0,\) and \(\xi( \cdot;\lambda) \) is a small part (it tends to zero at infinity).
The authors find the sets of parameter values for which the large-amplitude \(n\)-periodic trajectories exist for a fixed \(n.\) They show that in the related problems on small periodic orbits near zero, Arnold tongues are more narrow.

MSC:

39A11 Stability of difference equations (MSC2000)
37G10 Bifurcations of singular points in dynamical systems
39A12 Discrete version of topics in analysis
37C27 Periodic orbits of vector fields and flows
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