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Generalized Hopf bifurcations: a homological analysis and obstructions to their existence

  • Héctor Barge
  • J. J. Sánchez-Gabites
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Abstract

A generalized Hopf bifurcation takes place when varying a parameter of a (semi)dynamical system causes an attractor K to become a repeller. In doing so, it expels a (parameter-dependent) family of attractors. In this paper we concentrate on two questions: (1) what do the expelled attractors look like?, and (2) can any attractor K undergo a generalized Hopf bifurcation? Regarding (1), we compute the cohomology of the expelled attractors in terms of the cohomology of K; in the case of continuous-time dynamics this can be improved to a computation of their Borsuk shape. As for (2), we show that certain compacta cannot undergo generalized Hopf bifurcations and, when the ambient space has dimension $ \leq 3$, give a complete topological characterization of which can.

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Cite

BibTeXRIS
BibTeX
@misc{barge2026generalized,
  title = {{Generalized Hopf bifurcations: a homological analysis and obstructions to their existence}},
  author = {Héctor Barge and J. J. Sánchez-Gabites},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08002},
  eprint = {2611.08002},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08002},
}
RIS
TY  - UNPB
TI  - Generalized Hopf bifurcations: a homological analysis and obstructions to their existence
AU  - Héctor Barge
AU  - J. J. Sánchez-Gabites
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08002
UR  - https://arxiv.org/abs/2611.08002
AB  - A generalized Hopf bifurcation takes place when varying a parameter of a (semi)dynamical system causes an attractor K to become a repeller. In doing so, it expels a (parameter-dependent) family of attractors. In this paper we concentrate on two questions: (1) what do the expelled attractors look like?, and (2) can any attractor K undergo a generalized Hopf bifurcation? Regarding (1), we compute the cohomology of the expelled attractors in terms of the cohomology of K; in the case of continuous-time dynamics this can be improved to a computation of their Borsuk shape. As for (2), we show that certain compacta cannot undergo generalized Hopf bifurcations and, when the ambient space has dimension $ \leq 3$, give a complete topological characterization of which can.
ER  -