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 ≈ Cn < 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.