The Shape of Analyticity Domains of Lindstedt Series: the Standard Map
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چکیده
The analyticity domains of the Lindstedt series for the standard map are studied numerically using Padé approximants to model their natural boundaries. We show that if the rotation number is a Diophantine number close to a rational value p/q, then the radius of convergence of the Lindstedt series becomes smaller than the critical threshold for the corresponding KAM curve, and the natural boundary on the plane of the complexified perturbative parameter acquires a flower-like shape with 2q petals. (PACS Classification scheme: 05.45-a, 05.10-a, 45.10.Jj, 45.50.Pk) The standard map is a paradigmatic model for the transition from regular to stochastic motion in classical mechanics introduced by Chirikov [1]. It has also been studied in relation to problems of quantum mechanics and quantum chaos [2, 3], and statistical mechanics [4]; it is also relevant to problems in plasma physics [5]. It is a discrete one-dimensional dynamical system generated by the iteration of the symplectic map of the cylinder into itself, Tε : T×R 7→ T×R, given by Tε : x ′ = x+ y + ε sin x, y = y + ε sin x. For some background information, we refer the reader to the enormous literature on the topic, and in particular to [6] for a review. For ε = 0, the circles y = const. are invariant curves on which the dynamics is given by rotation with angular velocity – rotation number – ω = y/2π. As the perturbation is turned on, we face the classical KAM problem of determining which invariant curves survive and up to which size of the perturbative parameter ε (see [7, 8] for the optimal arithmetic condition on the rotation number for the stability of an invariant curve). It is well known that such invariant curves are given parametrically by the equation Cε,ω : {x = α+ u(α, ε, ω), y = 2πω + u(α, ε, ω) − u(α− 2πω, ε, ω)},
منابع مشابه
Scaling, Perturbative Renormalization and Analyticity for the Standard Map and Some Generalizations
We summarize some numerical works on the analyticity properties of the perturbative series (e. g. Lindstedt series) for some simple discrete-time hamiltonian systems like the so-called \standard map" and generalizations. We stress the relevance of our results for the development of a renormalization group picture based on perturbation theory.
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تاریخ انتشار 2000