The 18-cycle in Bianchi $VI_{-1/9}^{^{*}}$: Combined Linear Local Passage and Numerical Simulation
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912258396782592 |
|---|---|
| author | Buchner, Johannes |
| author_facet | Buchner, Johannes |
| contents | In this paper, we find an example for a periodic heteroclinic chain in Bianchi $VI_{-1/9}^{^{*}}$ that allows Takens Linearization at all base points. It turns out to be a "18-cycle'', i.e. involving a heteroclinic chain of 18 different base points at the Kasner circle. We then show that the combined cinear local passage at the 18-cycle is a contraction. This qualifies the 18-cycle as a candidate for proving the first rigorous convergence theorem in Bianchi $VI_{-1/9}^{^{*}}$.
For the numerical simulation of solutions that follow such heteroclinic cycles, we use a variable-step, variable-order (VSVO) Adams-Bashforth-Moulton PECE solver in Matlab. We conclude with a discussion on how to proceed further in studying Bianchi cosmologies, and also discuss directions for future research in inhomogeneous (PDE-) cosmological models. This puts our results in a broader perspective. The appendix contains symbolic and computations done by Mathematica for examples discussed throughout the text. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02684 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The 18-cycle in Bianchi $VI_{-1/9}^{^{*}}$: Combined Linear Local Passage and Numerical Simulation Buchner, Johannes Dynamical Systems Numerical Analysis General Relativity and Quantum Cosmology Mathematical Physics In this paper, we find an example for a periodic heteroclinic chain in Bianchi $VI_{-1/9}^{^{*}}$ that allows Takens Linearization at all base points. It turns out to be a "18-cycle'', i.e. involving a heteroclinic chain of 18 different base points at the Kasner circle. We then show that the combined cinear local passage at the 18-cycle is a contraction. This qualifies the 18-cycle as a candidate for proving the first rigorous convergence theorem in Bianchi $VI_{-1/9}^{^{*}}$. For the numerical simulation of solutions that follow such heteroclinic cycles, we use a variable-step, variable-order (VSVO) Adams-Bashforth-Moulton PECE solver in Matlab. We conclude with a discussion on how to proceed further in studying Bianchi cosmologies, and also discuss directions for future research in inhomogeneous (PDE-) cosmological models. This puts our results in a broader perspective. The appendix contains symbolic and computations done by Mathematica for examples discussed throughout the text. |
| title | The 18-cycle in Bianchi $VI_{-1/9}^{^{*}}$: Combined Linear Local Passage and Numerical Simulation |
| topic | Dynamical Systems Numerical Analysis General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2503.02684 |