🌊 Chaotic Waves — Nonlinear Oscillator Chains
A chain of N masses connected by nonlinear springs. In a purely harmonic chain energy stays in the initially excited mode forever (normal modes don't interact). Adding a nonlinear (anharmonic) coupling term makes the modes exchange energy — and at strong nonlinearity the dynamics becomes chaotic. This is the famous Fermi–Pasta–Ulam–Tsingou (FPUT) problem from 1953. The top canvas shows particle displacements; the bottom canvas shows the power spectrum of each normal mode.
Model
Parameters
Control
Stats
β-FPUT: V = ½x² + ¼βx⁴
Toda: V = (e^(αx)−αx−1)/α²
Equations of motion:
mẍᵢ = V′(xᵢ₊₁−xᵢ) − V′(xᵢ−xᵢ₋₁)
The FPUT Problem
In 1953 Fermi, Pasta, Ulam, and Tsingou simulated a chain of 64 particles with a small cubic nonlinearity. They expected energy to quickly thermalize across all modes — but instead observed near-recurrence: energy returned almost entirely to the first mode periodically. This violated the assumption of rapid equipartition of energy and puzzled physicists for years. The resolution involves solitons (Zabusky & Kruskal, 1965) and the KAM theorem. At strong nonlinearity the recurrence is destroyed and genuine chaos and thermalization occur.