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Type: Artigo de periódico
Title: A universal constant for semistable limit cycles
Author: Artés, Joan C.
Llibre, Jaume
Teixeira, Marco Antonio
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, and two limit cycles one stable and the other unstable if µ &gt; 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 &#8776; Cn&#945;< 0 with C, &#945; &#8712; R, such that the orbit of Xµn through a point of p &#8712; R reaches the position of the semistable limit cycle of X0 after given n turns. The exponent &#945; of this power law depends only on the multiplicity of the semistable limit cycle, and is independent of the initial point p &#8712; R and of the family Xµ. In fact &#945; = -2m/(2m - 1). Moreover the constant C is independent of the initial point p &#8712; R, but it depends on the family Xµ and on the multiplicity 2m of the limit cycle &#915;.
Subject: semistable limit cycle
semistable fixed point
universal constant
power law
Editor: Sociedade Brasileira de Matemática Aplicada e Computacional
Citation: Computational & Applied Mathematics. Sociedade Brasileira de Matemática Aplicada e Computacional, v. 30, n. 2, p. 463-483, 2011.
Rights: aberto
Identifier DOI: 10.1590/S1807-03022011000200012
Date Issue: 1-Jan-2011
Appears in Collections:Unicamp - Artigos e Outros Documentos

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