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We consider a planar four-vortex system with unit intensities and apply the KAM theorem for two-dimensional tori with fixed frequency. We obtain a rigorous lower bound for the stochasticity threshold of the torus with rotation number=(5—1)/2 and compare our result with numerical experiments. 相似文献
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A general direct method, alternative to KAM theory, apt to deal with small divisor problems in the real-analytic category, is presented and tested on several small divisor problems including the construction of maximal quasi-periodic solutions for nearly-integrable non-degenerate Hamiltonian or Lagrangian systems and the construction of lower dimensional resonant tori for nearly-integrable Hamiltonian systems. The method is based on an explicit graph theoretical representation of the formal power series solutions, which allows to prove compensations among the monomials forming such representation.L.C. thanks C. Simó and theCentre de Recerca Matemàtica (Bellaterra) for kind hospitality; he also acknowledges partial support by CNR-GNAFA. The authors gratefully acknowledge helpful discussions with C. Liverani. 相似文献
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We study numerically the complex domains of validity for KAM theory in generalized standard mappings. We compare methods based on Padé approximants and methods based on the study of periodic orbits. 相似文献
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A. Berretti A. Celletti L. Chierchia C. Falcolini 《Journal of statistical physics》1992,66(5-6):1613-1630
We consider KAM invariant curves for generalizations of the standard map of the form (x, y)=(x+y, y+f(x)), wheref(x) is an odd trigonometric polynomial. We study numerically their analytic properties by a Padé approximant method applied to the function which conjugates the dynamics to a rotation +. In the complex plane, natural boundaries of different shapes are found. In the complex plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to 0 as tends to the critical value. 相似文献
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L. Chierchia C. Falcolini 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1996,47(2):210-220
We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems u=f
x
(u, y), whereu y
M
u(y)
N
,f=f(x, y) is a real-analytic periodic function and is a small parameter. Kolmogorov's theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusT
M
+1. In the autonomous case,f=f(x), the above result holds for any . 相似文献
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We prove several theorems that lend support to Greene's criterion for the existence or not of invariant circles in twist maps. In particular, we show that some of the implications of the criterion are correct when the Aubry-Mather sets are smooth invariant circles or uniformly hyperbolic. We also suggest a simple modification that can work in the case that the Aubry-Mather sets have nonzero Lyapunov exponents. The latter is based on a closing lemma for sets with nonzero Lyapunov exponents, which may have several other applications. 相似文献
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