A universal constant for semistable limit cycles

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.