A Flying Lorenz Butterfly

A simple autonomous four-dimensional quadratic system is presented in which a two-wing chaotic attractor of Lorenz-type translates chaotically through phase space. The translation is generated by a single slow variable that functions as a dynamical offset. Continuous variation of the offset produces a visual impression of wing flapping. The system contains only five terms and two free parameters, remains fully quadratic, and preserves a clear butterfly structure while the center of the attractor executes a large-amplitude chaotic oscillation. The roles of the two parameters, the consequences of making them too large or too small, and a simple way to enhance the flapping are discussed. Numerical evidence of chaos is given.