How it works: The Duffing oscillator is a nonlinear ODE — ẍ + δẋ + αx + βx³ = γ cos(ωt). With α < 0 and β > 0 it forms a double-well potential with two stable equilibria. Periodic forcing can push the system into period doubling and eventually chaotic motion (positive Lyapunov exponent). The Poincaré section (right, red dots) samples the phase space once per drive period T = 2π/ω — a single dot means periodic motion; a fractal cloud means chaos.

Left panel: Phase portrait (x vs ẋ) with fading trail.   Right panel: Time series x(t) with Poincaré dots overlaid.