A universal constant for semistable limit cycles
Artés Ferragud, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Teixeira, Marco Antonio (Universidade Estadual de Campinas (Brasil). Departamento de Matemática)

Data:	2011
Resum:	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 ' 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.
Ajuts:	Ministerio de Economía y Competitividad MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-410
Nota:	Agraïments: The third author is partially supported by a grant FAPESP-2007/06896-5. All authors are also supported by the joint project CAPES-MECD grant HBP-2009-0025-PC.
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Matèria:	Semistable limit cycle ; Semistable fixed point ; Universal constant ; Power law
Publicat a:	Computational & Applied Mathematics, Vol. 30 Núm. 2 (2011) , p. 463-483, ISSN 1807-0302

DOI: 10.1590/S1807-03022011000200012

Postprint 19 p, 376.3 KB

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