Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

Abstract

Summary. In this paper we study the exponentially small splitting of a heteroclinic connection in a one-parameter family of analytic vector fields in R 3 . This family arises from the conservative analytic unfoldings of the so-called Hopf zero singularity (central singularity). The family under consideration can be seen as a small perturbation of an integrable vector field having a heteroclinic orbit between two critical points along the z axis. We prove that, generically, when the whole family is considered, this heteroclinic connection is destroyed. Moreover, we give an asymptotic formula of the distance between the stable and unstable manifolds when they meet the plane z = 0. This distance is exponentially small with respect to the unfolding parameter, and the main term is a suitable version of the Melnikov integral given in terms of the Borel transform of some function depending on the higher-order terms of the family. The results are obtained in a perturbative setting that does not cover the generic unfoldings of the Hopf singularity, which can be obtained as a singular limit of the considered family. To deal with this singular case, other techniques are needed. The reason to study the breakdown of the heteroclinic orbit is that it can lead to the birth of some homoclinic connection to one of the critical points in the unfoldings of the Hopf-zero singularity, producing what is known as a Shilnikov bifurcation.

The inner equation for generic analytic unfoldings of the Hopf-zero singularity

“…Let us observe that, taking ε = δ p+2 we recover the result given in [4] for p > −2 when ε is small and we deal with the case corresponding to a generic unfolding taking ε = 1.…”

“…Then the system becomes: , p = −2, and f, g, h = O( (δx, δy, δz, δ) 3 ) are analytic functions in B(r 0 ). In [4] the authors studied this system in the perturbative case p > −2 and they gave a rigorous proof of the breakdown of a heteroclinic orbit (located at x = y = 0) that exists if we consider only the terms coming from the normal form, that is, the case f = g = h = 0. The proof consisted in validating that the first order perturbation theory, that in this case was explicitly given by a Melnikov function, provided the correct prediction even if the Melnikov function (and the corresponding distance between the invariant manifolds) was exponentially small with respect to the parameter δ.…”

“…In [4], an asymptotic formula for the distance between W u and W s when they encounter the plane z = 0 was obtained when p > −2. This formula showed that this distance is exponentially small, in fact it is O(δ p e −π|α|/(2δ) ).…”

On the integrability and the zero-Hopf bifurcation of a Chen–Wang differential system

“…This theory of zero-Hopf bifurcation has been analyzed by Guckenheimer, Han, Holmes, Kuznetsov, Marsden and Scheurle in [8,9,13,14,17]. In particular they shown that some complicated invariant sets of the unfolding could bifurcate from the isolated zero-Hopf equilibrium under convenient conditions, showing that in some cases the zero-Hopf bifurcation could imply a local birth of "chaos", see for instance the articles [2,3,4,7,17] of Baldomá and Seara, Broer and Vegter, Champneys and Kirk, Scheurle and Marsden. Note that the differential system (1) only depends on one parameter so it cannot exhibit a complete unfolding of a zero-Hopf bifurcation. For studying the zero-Hopf bifurcation of system (1) we shall use the averaging theory in a similar way at it was used in [5] by Castellanos, Llibre and Quilantán.…”

Simultaneous Periodic Orbits Bifurcating from Two Zero-Hopf Equilibria in a Tritrophic Food Chain Model

“…In particular it is shown that some complicated invariant sets of the unfolding could bifurcate from the isolated zero-Hopf equilibrium under some conditions. Hence in some cases the zero-Hopf bifurcation could imply a local birth of "chaos" see for instance the articles [22][23][24][25][26] of Baldomá and Seara, Broer and Vegter, Champneys and Kirk, Scheurle and Marsden.…”