A universal constant for semistable limit cycles

IF 2.5 3区 数学 Q1 MATHEMATICS, APPLIED Computational & Applied Mathematics Pub Date : 2011-07-18 DOI:10.1590/S1807-03022011000200012
Joan C. Artés, J. Llibre, M. Teixeira
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Abstract

We consider one-parameter families of 2-dimensional vector fields Xµ having in a convenient region R a semistable limit cycle of multiplicity 2m when µ = 0, no limit cycles if µ 0. We show, analytically for some particular families and numerically for others, that associated to the semistable limit cycle and for positive integers n sufficiently large there is a power law in the parameter µ of the form µn ≈ Cnα< 0 with C, α ∈ R, such that the orbit of Xµn through a point of p ∈ R reaches the position of the semistable limit cycle of X0 after given n turns. The exponent α of this power law depends only on the multiplicity of the semistable limit cycle, and is independent of the initial point p ∈ R and of the family Xµ. In fact α = -2m/(2m - 1). Moreover the constant C is independent of the initial point p ∈ R, but it depends on the family Xµ and on the multiplicity 2m of the limit cycle Γ.
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半稳定极限环的一个通用常数
考虑二维向量场X μ的单参数族,当μ = 0时,在方便区域R中有一个多重2m的半稳定极限环,当μ = 0时没有极限环。我们用解析法和数值法证明了对于半稳定极限环和对于足够大的正整数n,在参数μ中存在一个幂律,其形式为μ n≈Cnα< 0,且C, α∈R,使得X μ n的轨道经过p∈R的一点,在给定n圈后到达X0的半稳定极限环的位置。幂律的指数α只依赖于半稳定极限环的多重性,与初始点p∈R和族Xµ无关。事实上,α = -2m/(2m - 1)。此外,常数C与初始点p∈R无关,但它取决于族Xµ和极限环的多重性2m Γ。
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来源期刊
Computational & Applied Mathematics
Computational & Applied Mathematics Mathematics-Computational Mathematics
CiteScore
4.50
自引率
11.50%
发文量
352
审稿时长
>12 weeks
期刊介绍: Computational & Applied Mathematics began to be published in 1981. This journal was conceived as the main scientific publication of SBMAC (Brazilian Society of Computational and Applied Mathematics). The objective of the journal is the publication of original research in Applied and Computational Mathematics, with interfaces in Physics, Engineering, Chemistry, Biology, Operations Research, Statistics, Social Sciences and Economy. The journal has the usual quality standards of scientific international journals and we aim high level of contributions in terms of originality, depth and relevance.
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