Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911387770421248 |
|---|---|
| author | Li, Dongchen Li, Xiaolong Shinohara, Katsutoshi Turaev, Dmitry |
| author_facet | Li, Dongchen Li, Xiaolong Shinohara, Katsutoshi Turaev, Dmitry |
| contents | We establish a necessary and sufficient condition for the birth of heterodimensional cycles from a generic homoclinic tangency to a hyperbolic periodic orbit. We prove for $C^r$ ($r=3,\dots,\infty,ω$) dynamical systems on a manifold $\mathcal{M}$, with $\dim \mathcal{M}\geqslant 3$ for diffeomorphisms and with $\dim \mathcal{M}\geqslant 4$ for flows, that $C^1$-robust heterodimensional dynamics of coindex one appear in any generic two-parameter $C^r$ unfolding of a homoclinic tangency to a periodic orbit such that at least one central multiplier is not real and the central dynamics are not sectionally dissipative. The heterodimensional dynamics also involve a blender exhibiting $C^1$-robust homoclinic tangencies. As a corollary, any system with a homoclinic tangency of the class described above belongs to the $C^r$ closure of the $C^1$-open Newhouse domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_14075 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies Li, Dongchen Li, Xiaolong Shinohara, Katsutoshi Turaev, Dmitry Dynamical Systems 37C05, 37C20, 37C25, 37C29 We establish a necessary and sufficient condition for the birth of heterodimensional cycles from a generic homoclinic tangency to a hyperbolic periodic orbit. We prove for $C^r$ ($r=3,\dots,\infty,ω$) dynamical systems on a manifold $\mathcal{M}$, with $\dim \mathcal{M}\geqslant 3$ for diffeomorphisms and with $\dim \mathcal{M}\geqslant 4$ for flows, that $C^1$-robust heterodimensional dynamics of coindex one appear in any generic two-parameter $C^r$ unfolding of a homoclinic tangency to a periodic orbit such that at least one central multiplier is not real and the central dynamics are not sectionally dissipative. The heterodimensional dynamics also involve a blender exhibiting $C^1$-robust homoclinic tangencies. As a corollary, any system with a homoclinic tangency of the class described above belongs to the $C^r$ closure of the $C^1$-open Newhouse domain. |
| title | Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies |
| topic | Dynamical Systems 37C05, 37C20, 37C25, 37C29 |
| url | https://arxiv.org/abs/2203.14075 |