Duffing Oscillator
The Duffing equation describes a mass on a nonlinear spring with a cubic restoring force, subject to damping and periodic forcing. Despite its simple form, it is one of the canonical routes to chaos in dynamical systems — displaying period-doubling cascades, coexisting attractors, and fractal basin boundaries. Originally introduced by Georg Duffing in 1918 to model hard-spring mechanical systems.
ẍ + δẋ + αx + βx³ = γ cos(ωt) (Duffing equation)
Switch views above. Poincaré section samples phase space once per drive period — chaos shows fractal structure. Bifurcation sweeps γ to reveal the period-doubling route to chaos.
The Nonlinear Spring
A linear spring obeys F = −αx exactly. A Duffing spring adds a cubic term: F = −αx − βx³. When β > 0 the spring stiffens as it stretches (hard spring); when β < 0 it softens (soft spring) — potentially giving two stable wells separated by an unstable equilibrium at the origin (double-well potential).
Under periodic forcing with frequency ω and amplitude γ, the response can be: (1) periodic with the drive, (2) subharmonic at ω/2, ω/3..., (3) quasiperiodic, or (4) fully chaotic. The transition from periodic to chaotic passes through a period-doubling cascade whose accumulation point is a universal constant (Feigenbaum's δ ≈ 4.669).
- Phase portrait: closed loops for periodic orbits, dense winding curves for chaos.
- Poincaré section: a single point for period-1, N points for period-N, fractal set for chaos.
- Bifurcation diagram: sweeps drive amplitude γ; branching reveals period doublings and chaos windows.
- Coexisting attractors: in the double-well case, two stable orbits can coexist — history determines which basin the trajectory occupies.