On the relationship between the Bruno function and the breakdown of invariant tori

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• 作者：Locatelli ; Ugo ; Froeschle´ ; Claude ; Lega ; Elena ; Morbidelli ; Alessandro
• 关键词：Bruno function ; Quasiperiodic motions ; Number theory ; Hamiltonian systems
• 刊名：Physica D
• 出版年：2000
• 出版时间：May 1, 2000
• 年：2000
• 卷：139
• 期：1-2
• 页码：48-71
• 全文大小：341 K

文摘

We study the ratio _c(ω)/exp(−ηB(ω)), where _c(ω) is the breakdown threshold function for an analytic invariant torus, η is a parameter and B(ω) is the Bruno function, which is purely arithmetic (i.e., it only depends on the number theory properties of ω). We consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with Diophantine rotation frequency. Our numerical experiments, together with some heuristic considerations, strongly suggest that _c(ω)/exp(−ηB(ω)) is not a continuous function on Diophantine numbers ω, for all values of η.