Analytic proof of the emergence of new type of Lorenz-like attractors from the triple instability in systems with $\mathbb{Z}_4$-symmetry

Efrosiniia Karatetskaia; Alexey Kazakov; Klim Safonov; Dmitry Turaev

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arXiv · 2408.06066

Analytic proof of the emergence of new type of Lorenz-like attractors from the triple instability in systems with $\mathbb{Z}_4$-symmetry

Efrosiniia Karatetskaia·Alexey Kazakov·Klim Safonov·Dmitry Turaev

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Abstract

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 Sim\'o 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 Sim\'o angels in the three-dimensional H\'enon map.

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Efrosiniia Karatetskaia, Alexey Kazakov, Klim Safonov, Dmitry Turaev. 2024-08-12. Analytic proof of the emergence of new type of Lorenz-like attractors from the triple instability in systems with $\mathbb{Z}_4$-symmetry. https://arxiv.org/abs/2408.06066

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