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

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.