Heterodimensional cycles derived from homoclinic tangencies via Hopf bifurcations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916742692864000 |
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| author | Tomizawa, Shuntaro |
| author_facet | Tomizawa, Shuntaro |
| contents | We analyze three-dimensional $C^{r}$ diffeomorphisms ($r\ge 5$) exhibiting a quadratic focus-saddle homoclinic tangency whose multipliers satisfy $|λγ| = 1$. For a proper three-parameter unfolding that splits the tangency, varies the argument of the stable multipliers, and controls the modulus $|λγ|$, we show that a Hopf bifurcation occurs on this curve and that a homoclinic point to the bifurcating periodic orbit is present. As a consequence, the original map $f$ can be $C^{r}$-approximated by a diffeomorphism exhibiting a coindex-one heterodimensional cycle in the saddle case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12596 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heterodimensional cycles derived from homoclinic tangencies via Hopf bifurcations Tomizawa, Shuntaro Dynamical Systems 37G25, 37C29, 37G15, 37D30, 37C20 We analyze three-dimensional $C^{r}$ diffeomorphisms ($r\ge 5$) exhibiting a quadratic focus-saddle homoclinic tangency whose multipliers satisfy $|λγ| = 1$. For a proper three-parameter unfolding that splits the tangency, varies the argument of the stable multipliers, and controls the modulus $|λγ|$, we show that a Hopf bifurcation occurs on this curve and that a homoclinic point to the bifurcating periodic orbit is present. As a consequence, the original map $f$ can be $C^{r}$-approximated by a diffeomorphism exhibiting a coindex-one heterodimensional cycle in the saddle case. |
| title | Heterodimensional cycles derived from homoclinic tangencies via Hopf bifurcations |
| topic | Dynamical Systems 37G25, 37C29, 37G15, 37D30, 37C20 |
| url | https://arxiv.org/abs/2505.12596 |