Conjoined Lorenz twins—a new pseudohyperbolic attractor in three-dimensional maps and flows

We describe new types of Lorenz-like attractors for three-dimensional flows and maps with symmetries. We give an example of a three-dimensional system of differential equations, which is centrally symmetric and mirror symmetric. We show that the system has a Lorenz-like attractor, which contains three saddle equilibrium states and consists of two mirror-symmetric components that are adjacent at the symmetry plane. We also found a discrete-time analog of this “conjoined-twins” attractor in a cubic three-dimensional Hénon map with a central symmetry. We show numerically that both attractors are pseudohyperbolic, which guarantees that each orbit of the attractor has a positive maximal Lyapunov exponent, and this property is preserved under small perturbations. We also describe bifurcation scenarios for the emergence of the attractors in one-parameter families of three-dimensional flows and maps possessing the symmetries.