Home > Articles > Published articles > A universal constant for semistable limit cycles |
Date: | 2011 |
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 ' 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. |
Grants: | Ministerio de Economía y Competitividad MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-410 |
Note: | 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. |
Rights: | Tots els drets reservats. |
Language: | Anglès |
Document: | Article ; recerca ; Versió acceptada per publicar |
Subject: | Semistable limit cycle ; Semistable fixed point ; Universal constant ; Power law |
Published in: | Computational & Applied Mathematics, Vol. 30 Núm. 2 (2011) , p. 463-483, ISSN 1807-0302 |
Postprint 19 p, 376.3 KB |