〰️ Nonlinear Oscillations — Duffing Oscillator
The Duffing oscillator ẍ + δẋ + αx + βx³ = γcos(ωt) is the canonical model of nonlinear resonance and deterministic chaos. Vary the forcing amplitude γ and watch the system transit from periodic motion through period-doubling to chaos. The phase portrait shows trajectories; the Poincaré section samples one point per period; the bifurcation diagram records long-run attractors vs. γ.
Parameters
Presets
State
ẍ + δẋ + αx + βx³ = γcos(ωt)
V(x) = ½αx² + ¼βx⁴
Two wells when α<0, β>0
About Nonlinear Oscillations
Unlike a linear harmonic oscillator, the Duffing system with a double-well potential (α < 0, β > 0) can exhibit sensitive dependence on initial conditions — the hallmark of deterministic chaos discovered by Edward Lorenz and studied rigorously in the nonlinear dynamics work of Ueda, Guckenheimer, and Holmes. The Poincaré section stroboscopically samples the state (x, ẋ) once per forcing period; for periodic orbits this gives finitely many points, for chaos a fractal strange attractor. The bifurcation diagram sweeps γ and records the long-run Poincaré x-coordinates.