Mathematics and beauty: Time-discrete phase planes associated with the cyclic system, { x ̇(t) = -f(y(t)), ẏ(t) = f(x(t))}

Abstract

Portraits of time-discrete phase planes associated with the cyclic system [ x ̇(t) = -f(y(t)), y ̇(t) = f(x(t))} are presented, where f is a frequency modulation term of the general form, f(x) = sin[x + sin(ρ{variant}x)]. The resulting trajectories are of interest artistically and mathematically, and they reveal a visually striking and intricate class of patterns ranging from stable points to hierarchies of stable cycles and to apparently random fluctuations. The computer-based system presented is special in its primary focus on the fast characterization of simple cyclic systems using an interactive graphics system with a variety of controlling parameters. © 1987.