Analytic proof of the emergence of new type of Lorenz-like attractors from the triple instability in systems with $\mathbb{Z}_4$-symmetry
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914909863804928 |
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| author | Karatetskaia, Efrosiniia Kazakov, Alexey Safonov, Klim Turaev, Dmitry |
| author_facet | Karatetskaia, Efrosiniia Kazakov, Alexey Safonov, Klim Turaev, Dmitry |
| contents | We study bifurcations of a symmetric equilibrium state in systems of differential equations invariant with respect to a $\mathbb{Z}_4$-symmetry. We prove that if the equilibrium state has a triple zero eigenvalue, then pseudohyperbolic attractors of different types can arise as a result of the bifurcation. The first type is the classical Lorenz attractor and the second type is the so-called Simó angel. The normal form of the considered bifurcation also serves as the normal form of the bifurcation of a periodic orbit with multipliers $(-1, i, -i)$. Therefore, the results of this paper can also be used to prove the emergence of discrete pseudohyperbolic attractors as a result of this codimension-3 bifurcation, providing a theoretical confirmation for the numerically observed Lorenz-like attractors and Simó angels in the three-dimensional Hénon map. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_06066 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analytic proof of the emergence of new type of Lorenz-like attractors from the triple instability in systems with $\mathbb{Z}_4$-symmetry Karatetskaia, Efrosiniia Kazakov, Alexey Safonov, Klim Turaev, Dmitry Dynamical Systems We study bifurcations of a symmetric equilibrium state in systems of differential equations invariant with respect to a $\mathbb{Z}_4$-symmetry. We prove that if the equilibrium state has a triple zero eigenvalue, then pseudohyperbolic attractors of different types can arise as a result of the bifurcation. The first type is the classical Lorenz attractor and the second type is the so-called Simó angel. The normal form of the considered bifurcation also serves as the normal form of the bifurcation of a periodic orbit with multipliers $(-1, i, -i)$. Therefore, the results of this paper can also be used to prove the emergence of discrete pseudohyperbolic attractors as a result of this codimension-3 bifurcation, providing a theoretical confirmation for the numerically observed Lorenz-like attractors and Simó angels in the three-dimensional Hénon map. |
| title | Analytic proof of the emergence of new type of Lorenz-like attractors from the triple instability in systems with $\mathbb{Z}_4$-symmetry |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2408.06066 |