Listening for chaos

Strike it, watch and hear the motion turn regular as the energy decays; Lyapunov exponent and Poincaré section.

A struck chain of N masses with cubic springs; energy spreading up the modes, Lyapunov exponent and Poincaré sections for N = 2 and 3.

A struck beam of N masses with bending stiffness and stretching: clamped, pinned or cantilever; inharmonic, cymbal-like overtones.

A struck square plate of N × N masses with bending stiffness and von Kármán stretching, simply supported, clamped or free; modes (m, n) at m² + n², pitch glides when struck hard.

A rectangular plate driven in one mode, with no damping, its energy raised slowly: watch the energy move from the driven mode into the others.