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Title:
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Hopf-Zero singularities truly unfold chaos
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Author:
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Baldomá, I.; Ibáñez, S.; Seara, T.M.
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Abstract:
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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 singulari-
ties and generic unfoldings. The existence of Shilnikov bifurcations in the C r 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 conditions only depend on the 2-jet of the family. |
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Publication date:
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2019-12-30 |
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Subject (UDC):
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51 - Matemàtiques |
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Subject(s):
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Matemàtiques |
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Rights:
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L'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons:http://creativecommons.org/licenses/by-nc-sa/4.0/ |
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Pages:
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20 p. |
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Document type:
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Article
Article - Published version |
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DOI:
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10.1016/j.cnsns.2019.105162
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Published by:
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Elsevier
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Publish at:
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Communications in Nonlinear Science and Numerical Simulation
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Share:
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