Strict increase in the number of normally hyperbolic limit tori in 3D polynomial vector fields
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915250053316608 |
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| author | Arakaki, Lucas Queiroz Novaes, Douglas D. |
| author_facet | Arakaki, Lucas Queiroz Novaes, Douglas D. |
| contents | The second part of Hilbert's 16th problem concerns determining the maximum number $H(m)$ of limit cycles that a planar polynomial vector field of degree $m$ can exhibit. A natural extension to the three-dimensional space is to study the maximum number $N(m)$ of limit tori that can occur in spatial polynomial vector fields of degree $m$. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number $N_h(m)$, if finite, increases strictly with $m$. More precisely, we prove that $N_h(m+1) \geqslant N_h(m) + 1$. Our proof relies on the torus bifurcation phenomenon observed in spatial vector fields near Hopf-Zero equilibria. While conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of a torus bifurcation assuming only that the linear part of the unperturbed vector field is in Jordan normal form. This approach circumvents the need for intricate computations involving higher-order normal forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13832 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strict increase in the number of normally hyperbolic limit tori in 3D polynomial vector fields Arakaki, Lucas Queiroz Novaes, Douglas D. Dynamical Systems Classical Analysis and ODEs 34C23, 34C29, 34C45 The second part of Hilbert's 16th problem concerns determining the maximum number $H(m)$ of limit cycles that a planar polynomial vector field of degree $m$ can exhibit. A natural extension to the three-dimensional space is to study the maximum number $N(m)$ of limit tori that can occur in spatial polynomial vector fields of degree $m$. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number $N_h(m)$, if finite, increases strictly with $m$. More precisely, we prove that $N_h(m+1) \geqslant N_h(m) + 1$. Our proof relies on the torus bifurcation phenomenon observed in spatial vector fields near Hopf-Zero equilibria. While conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of a torus bifurcation assuming only that the linear part of the unperturbed vector field is in Jordan normal form. This approach circumvents the need for intricate computations involving higher-order normal forms. |
| title | Strict increase in the number of normally hyperbolic limit tori in 3D polynomial vector fields |
| topic | Dynamical Systems Classical Analysis and ODEs 34C23, 34C29, 34C45 |
| url | https://arxiv.org/abs/2504.13832 |