A universal constant for semistable limit cycles

Abstract

Abstract. We consider one-parameter families of 2-dimensional vector fields X μ 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 μ n ≈ Cn α < 0 with C, α ∈ R, such that the orbit of X μ n through a point of p ∈ R reaches the position of the semistable limit cycle of X 0 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 X μ . In fact α = −2m/(2m − 1).Moreover the constant C is independent of the initial point p ∈ R, but it depends on the family X μ and on the multiplicity 2m of the limit cycle .Mathematical subject classification: 58F14, 58F21, 58F30.