Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity

Abstract

In this paper we prove the breakdown of an heteroclinic connection in the analytic versal unfoldings of the generic Hopf-Zero singularity in an open set of the parameter space. This heteroclinic orbit appears at any order if one performs the normal form around the origin, therefore it is a phenomenon "beyond all orders". In this paper we provide a formula for the distance between the corresponding stable and unstable one dimensional manifolds which is given by an exponentially small function in the perturbation parameter. Our result applies both for conservative and dissipative unfoldings.

Cited by in corpus (9)

• Zero-Hopf bifurcation in a Chua system
• Hopf-Zero singularities truly unfold chaos
• Bifurcation control and universal unfolding for Hopf-zero singularities with leading solenoidal terms
• Accelerator modes and anomalous diffusion in 3D volume-preserving maps
• Separatrix splitting at a Hamiltonian $0^{2}i\omega$ bifurcation
• Breakdown of heteroclinic connections in the analytic Hopf-Zero singularity: Rigorous computation of the Stokes constant
• Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (I)
• Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (II). The generic case
• On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting