Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity

Abstract

In this paper we study the exponentially small splitting of a heteroclinic orbit in some unfoldings of the central singularity also called Hopf-zero singularity. The fields under consideration are of the form: dx dτ = −δxz − y (α + cδz) + δp+1f(δx, δy, δz, δ) dy dτ = −δyz + x (α + cδz) + δp+1g(δx, δy, δz, δ) dz dτ = δ ?−1 + b(x2 + y2) + z2? + δp+1h(δx, δy, δz, δ), where f, g and h are real analytic functions, α, b and c are constants and δ is a small parameter. When f = g = h = 0 the system has a heteroclinic orbit between the critical points (0, 0,±1) given by: {(x, y) = (0, 0) ;−1 < z < 1}. Let ds,u be the distance between the one dimensional stable and unstable manifold of the perturbed system measured at the plane z = 0. We prove that for any f, g such that ˆm(i α) ?= 0, where ˆm is the Borel transform of the function m(u) = u1+i c(f + i g)(0, 0, u, 0) |ds,u| = 2π ecπ/2 | ˆm(i α)|δp e−π|α|/(2δ)(1 + O(δp+2| log δ|)), p>−2.