On Normal Forms of Reversible Systems with Lorenz Attractors and Repellers

We consider two-parameter families of three-dimensional systems, which are normal forms for bifurcations of an equilibrium state with three zero eigenvalues in the class of systems that are time-reversible and axially symmetric. We identify those normal forms in which bifurcations can be observed, leading to the emergence of symmetrical “Lorenz attractor – Lorenz repeller” pairs. We illustrate only four examples of such normal forms in which we describe bifurcation scenarios leading to the emergence of different types of such pairs.