A universal constant for semistable limit cycles

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 2 R{double struck}, 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. © 2011 SBMAC.