A point mass slides along the real-space potential curve V(x) = ½αx² + ¼βx⁴ under gravity-like restoring force, damping and a periodic drive. The equation of motion is the Duffing equation, integrated here with 4th-order Runge–Kutta (RK4) at a fixed 4 ms sub-step:
ẍ + δẋ + αx + βx³ = γ·cos(ωt)
V(x) = ½αx² + ¼βx⁴
When α<0 and β>0 the curve forms two wells with a hump between them — the mass can sit in either well or, once the forcing pumps in enough energy, hop back and forth between them. The translucent twin ball starts a hair's-width away in velocity; watching it peel away from the main ball over time is a direct, real-space view of the sensitive dependence on initial conditions that defines deterministic chaos.
- Main view — the potential curve V(x) with the ball's real position and a fading trail of its past positions.
- Phase inset — x vs. ẋ, the abstract trajectory the ball traces in state space.
- Twin separation — |Δx| + |Δẋ| between the main ball and its twin; growth here is the chaos signature.