FPUT Chain

Ideal β-FPUT chain with N point masses and spring force F(r) = r + βr³. Mass and linear stiffness are 1 in normalised units. Fixed ends include wall springs; free ends include a zero-frequency translation mode. The initial displacement excites mode 1 with zero velocity.

Recurrence transfers projected energy away from mode 1 and nearly returns it around t = 18 for the preset N = 4, β = 4, A = 2, γ = 0. This is not an exact return of the entire state. Linear retains the first mode; damping γ removes energy. Shares are normalised by the sum of harmonic modal energies, which excludes the nonlinear spring potential. The displayed total energy E includes it.

The vertical displacement is a signal plot, not a transverse mechanical model. Step advances by 0.005 and pauses; physical edits reset the model, while tempo changes preserve it. Hidden tools stop advancing. History retains up to 1200 samples across the complete time span. Zero amplitude has no defined shares. PNG saves the share plot with its legend.