Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity
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Cita com:
hdl:2117/887
Tipus de documentArticle
Data publicació2004
Condicions d'accésAccés obert
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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.
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