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On universality for area-preserving maps of the plane. (English) Zbl 1194.37050

Summary: We present detailed evidence that one-parameter families of area-preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter. We relate this behaviour to a fixed point equation of the form \(\Lambda^{ - 1}\circ\Phi\circ\Phi\circ\Lambda = \Phi \) and \(\det D\Phi = 1\), \(\Phi:\mathbb{R}^2\rightarrow\mathbb{R}^{2}\). In particular we argue that the scaling transformation \(\Lambda:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}\) is conjugate to the transformation \(\Lambda_{0}:(x, y)\rightarrow (\lambda x, \mu y)\), with \(\lambda ^{2} \neq \mu \), and in fact \(\lambda^{2} > \mu\). We present some numerical evidence that \(\delta = 8.721\dots\), \(- 1/\lambda = 4.018\dots\), \(1/\mu = 16.36\dots\), where \(\delta \) is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods \(2^{k}\) described above.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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References:

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