Flying Lorenz Butterfly

J. C. Sprott
Department of Physics
University of Wisconsin
Madison, WI 53706, USA
July 29, 2026

A simple autonomous four-dimensional quadratic system in which a two-wing Lorenz-like attractor slowly translates through phase space. The translation is produced by a dynamical offset variable u(t); the resulting chaotic motion of the center creates a visual impression of wing flapping. The system is

dx/dt = y - x
dy/dt = -x(z - u)
dz/dt = xy - 1
du/dt = δx - μu

with representative parameters δ = 0.05 and μ = 0.001. Color is used to indicate the local Lyapunov exponent, with red positive (expanding) and blue negative (contracting). The animation shows 50 frames at 5 frames per second and loops continuously, spanning approximately 6 x10⁵ time units. A short paper describing the system has been submitted for publication.