%0 Journal Article
%T A universal constant for semistable limit cycles
%A Art¨¦s
%A Joan C.
%A Llibre
%A Jaume
%A Teixeira
%A Marco Antonio
%J Computational & Applied Mathematics
%D 2011
%I Scientific Electronic Library Online
%R 10.1590/S1807-03022011000200012
%X we consider one-parameter families of 2-dimensional vector fields x¦Ì having in a convenient region r a semistable limit cycle of multiplicity 2m when ¦Ì = 0, no limit cycles if ¦Ì < 0, and two limit cycles one stable and the other unstable if ¦Ì > 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 ¦Ì of the form ¦Ìn ¡Ö cn¦Á< 0 with c, ¦Á ¡Ê r, such that the orbit of x¦Ìn through a point of p ¡Ê r reaches the position of the semistable limit cycle of x0 after given n turns. the exponent ¦Á of this power law depends only on the multiplicity of the semistable limit cycle, and is independent of the initial point p ¡Ê r and of the family x¦Ì. in fact ¦Á = -2m/(2m - 1). moreover the constant c is independent of the initial point p ¡Ê r, but it depends on the family x¦Ì and on the multiplicity 2m of the limit cycle ¦Ã.
%K semistable limit cycle
%K semistable fixed point
%K universal constant
%K power law.
%U http://www.scielo.br/scielo.php?script=sci_abstract&pid=S1807-03022011000200012&lng=en&nrm=iso&tlng=en