Hopf-Zero singularities truly unfold chaos

• Published in: Communications in nonlinear science & numerical simulation 2020-05, Vol.84, p.105162, Article 105162
• Main Authors: Baldomá, I., Ibáñez, S., Seara, T.M.
• Format: Article
• Language: English
• Subjects: Bifurcations; Canonical forms; Chaos theory; Mathematical analysis; Mathematical models; Numerical analysis; Singularities; Topological manifolds; Topology
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2019.105162

•This is one of the few works rigorously proving the existence of chaotic phenomena for three dimensional vector fields.•We prove that chaotic motions actually take place by means of proving the existence of homoclinic orbits of Shilnikov type. Such orbits arise from a beyond all orders phenomena: the exponentially small breakdown of certain heteroclinic invariant manifolds.•It is a well-known result that a system possessing a Shilnikov orbit has associated infinitely many Smale’s horseshoes and therefore, a hyperbolic invariant set over which the restricted dynamics is conjugated with the infinitely many symbols shift.•The vector fields under consideration are generic analytic unfoldings of a class of Hopf-Zero singularities. We provide computable conditions which ensure the existence of Shilnikov homoclinic orbits and, therefore, the existence of chaotic dynamics. We provide conditions to guarantee the occurrence of Shilnikov bifurcations in analytic unfoldings of some Hopf-Zero singularities through a beyond all order phenomenon: the exponentially small breakdown of invariant manifolds which coincide at any order of the normal form procedure. The conditions are computable and satisfied by generic singularities and generic unfoldings. The existence of Shilnikov bifurcations in the Cr case was already argued by Guckenheimer in the 80’s. About the same time, endowing the space of C∞ unfoldings with a convenient topology, persistence and density of the Shilnikov phenomenon was proved by Broer and Vegter in 1984. However, since the proof involves the use of flat perturbations, this approach is not valid in the analytic context. What is more, none of the mentioned approaches provides a computable criteria to decide whether a given unfolding exhibits Shilnikov bifurcations or not. Many people appeals to the appearance of Hopf-Zero singularities to explain the emergence of chaos in a huge number of applications. However, no one can refer to a specific theorem establishing the conditions that a given unfolding should satisfy to ensure that chaotic dynamics are exhibited. We fill this gap by providing an ultimate result about the appearance of Shilnikov bifurcations in analytic unfoldings of a certain class of Hopf-Zero singularities. These conditions are computable and satisfied by generic families. One of these conditions depends on the full jet of the singularity and comes from a beyond all order phenomenon. It can be related with Stokes constants. The other c