New Lorenz-like Chaotic Flows with Minimal Algebraic Structure

In 1994, a numerical examination by J.C. Sprott of three-dimensional autonomous ordinary differential equations with quadratic nonlinearities uncovered five new simple examples of chaotic flows with either of the five terms including two nonlincaritics on the right hand side of the differential equations. Two of these flows resemble the Lorenz system as they are cquivariam under the action of the rotation through π radians around the z-axis. Moreover, both exhibit two off-axis fixed points and two-scroll butterfly shaped chaotic attraelors. In canonical form, these systems have in the second equation the same nonlinearity xz, as that of the Lorenz system, and the nonlinearity in the third equation is xy or y². The present paper introduces the first known examples of Lorenz-like systems with the simplest algebraic structure including the nonlinearity x² in the third equation when written in canonical form, and allowing chaotic behaviour. Of particular interest is the fact that these new chaotic flows generate one-scroll non-butterfly shaped chaotic attmotors surrounding two off-axis equilibria points.