Three-dimensional horseshoes near an unfolding of a Hopf-Hopf singularity
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| Format: | Preprint |
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2025
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| _version_ | 1866909597809246208 |
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| author | Ibáñez, Santiago Rodrigues, Alexandre A. P. |
| author_facet | Ibáñez, Santiago Rodrigues, Alexandre A. P. |
| contents | Motivated by a certain type of unfolding of a Hopf-Hopf singularity, we consider a one-parameter family $(f_γ)_{γ\geq0}$ of $C^3$--vector fields in $\mathbb{R}^4$ whose flows exhibit a heteroclinic cycle associated to two periodic solutions and a bifocus, all of them hyperbolic. It is formally proved that combining rotation with a generic condition concerning the transverse intersection between the three-dimensional invariant manifolds of the periodic solutions, all sets are highly distorted by the first return map and hyperbolic three-dimensional horseshoes emerge, accumulating on the network. Infinitely many linked horseshoes prompt the coexistence of infinitely many saddle-type invariant sets for all values of $γ\gtrsim 0$ belonging to the heteroclinic class of the two hyperbolic periodic solutions. We apply the results to a particular unfolding of the Hopf-Hopf singularity, the so called \emph{Gaspard-type unfolding}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_21783 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Three-dimensional horseshoes near an unfolding of a Hopf-Hopf singularity Ibáñez, Santiago Rodrigues, Alexandre A. P. Dynamical Systems 34C37, 34D20, 37C27 Motivated by a certain type of unfolding of a Hopf-Hopf singularity, we consider a one-parameter family $(f_γ)_{γ\geq0}$ of $C^3$--vector fields in $\mathbb{R}^4$ whose flows exhibit a heteroclinic cycle associated to two periodic solutions and a bifocus, all of them hyperbolic. It is formally proved that combining rotation with a generic condition concerning the transverse intersection between the three-dimensional invariant manifolds of the periodic solutions, all sets are highly distorted by the first return map and hyperbolic three-dimensional horseshoes emerge, accumulating on the network. Infinitely many linked horseshoes prompt the coexistence of infinitely many saddle-type invariant sets for all values of $γ\gtrsim 0$ belonging to the heteroclinic class of the two hyperbolic periodic solutions. We apply the results to a particular unfolding of the Hopf-Hopf singularity, the so called \emph{Gaspard-type unfolding}. |
| title | Three-dimensional horseshoes near an unfolding of a Hopf-Hopf singularity |
| topic | Dynamical Systems 34C37, 34D20, 37C27 |
| url | https://arxiv.org/abs/2504.21783 |