About the unfolding of a Hopf-zero singularity

Freddy Dumortier
Universiteit Hasselt, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590,
Diepenbeek, Belgium
E-mail: freddy.dumortier@uhasselt.be

Santiago Ib\'a\~nez
Departamento de Matem\'aticas, Universidad de Oviedo, Avda. Calvo Sotelo s/n,
33007 Oviedo, Spain
E-mail: mesa@uniovi.es

Hiroshi Kokubu 
Department of Mathematics/JST-CREST, Kyoto University, Kyoto 606-8502, Japan
E-mail: kokubu@math.kyoto-u.ac.jp

Carles Sim\'o
Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona
Gran Via, 585, 08071 Barcelona, Spain
E-mail: carles@maia.ub.es

Abstract
We study arbitrary generic unfoldings of a Hopf-zero singularity of codimension
two. They can be written in the following
normal form:
\[ \left\{\begin{array}{l}
    x'=-y+\mu x-axz+A(x,y,z,\lambda,\mu) \\
    y'=x+\mu y-ayz+B(x,y,z,\lambda,\mu) \\
    z'=z^2+\lambda+b(x^2+y^2)+C(x,y,z,\lambda,\mu),
 \end{array} \right.
\]
where $A$, $B$, $C$ are $C^\infty$ or $C^\omega$ and of order
$O(\|(x,y,z,\lambda,\mu)\|^3)$. Moreover we focus on the most interesting case,
namely, the topological type of codimension two characterized by the open
conditions $a>0$ and $b>0$.

At second order the above family is integrable for $\mu=0$ and, when
$\lambda < 0$, there exist on the vertical axis two saddle-focus equilibrium
points with different stability indices. Their respective two-dimensional
invariant manifolds coincide and form an ellipsoid. Inside, the equilibrium
points are connected by a heteroclinic orbit along the vertical axis. Such
configuration can be destroyed by higher order terms and therefore homoclinic
and heteroclinic orbits can appear. The saddle-focus homoclinic orbits will
satisfy the Shilnikov conditions for $0<a<2$ and, in such a case, chaotic
dynamics should be present in the system for some parameter values.

The determinacy of the splitting function is a very involved problem which has
been tackled by several authors. The difficulty lies in the fact that some
higher order terms have an exponentially small effect in the splitting. Moreover
no formula, valid for any generic unfolding, for the coefficients in the leading
terms of the splitting function is available. In fact, unless we use the exact
form of the vector fields we cannot expect to get such a formula. No truncation
can provide information about exponentially small phenomena. A different
question is to know what happens in general, that is, the behaviour out of a set
of families of large codimension.

In this paper we work with meaningful expressions for the splitting functions.
They are supported by partial theoretical results present in the literature and
also by numerical results. We generically assume that the leading coefficients
do not vanish and explore the dynamical consequences of certain additional mild
conditions.

We will also pay attention to the so-called Michelson system, a specific
unfolding of the Hopf-zero singularity consisting of volume preserving vector
fields, and hence leading to measure preserving families of diffeomorphisms for
suitable Poincar\'e maps. Michelson system is related to the study of the
unfolding of the three-dimensional nilpotent singularity of codimension three
and as well as to the Kuramoto-Shivashinsky equation. Numerical results
supporting the validity of the splitting assumptions and showing several aspects
of the dynamics are also presented.