How is this different from the 3D version?

The 3D version renders abstract views only — phase portrait, Poincare section and bifurcation diagram over a shader background. This 2D companion instead draws the real-space potential curve V(x) and shows the mass physically rolling along it, with the phase portrait reduced to a small inset rather than the main view.

What does the twin ball show?

The twin ball starts with a velocity offset of just 0.02 from the main ball. Under chaotic parameters the two trajectories separate rapidly even though they started almost identical, which is a direct visualization of sensitive dependence on initial conditions.

Why does the ball sit in one of two wells?

When alpha is negative and beta is positive the potential V(x) = half alpha x squared plus a quarter beta x to the fourth has two minima separated by a hump. The ball naturally settles toward whichever minimum it is closer to, and periodic forcing can kick it across the hump into the other well.

Home▸Chaos & Dynamics▸Duffing Oscillator: Ball in a Double Well (2D)

Duffing Oscillator: Ball in a Double Well (2D)

2D Duffing-oscillator lab: watch a mass roll through a real-space double-well potential under periodic forcing, with a live phase-plot inset and a twin trajectory released a hair apart to reveal chaotic divergence.

Chaos & Dynamics2DModerate60 FPS📱 Mobile-adapted⇄ 3D version

This 2D companion drives the same Duffing equation ẍ + δẋ + αx + βx³ = γcos(ωt) as the 3D version, but renders it in real space rather than through abstract phase/Poincaré/bifurcation tabs: the main canvas draws the potential curve V(x) = ½αx² + ¼βx⁴ itself, and a ball physically rolls along it, hopping between the two wells whenever the periodic forcing pumps in enough energy. A translucent twin ball, released with a velocity offset of just 0.02 from the main one, visibly peels apart from it under chaotic parameters — making the sensitive dependence on initial conditions that defines chaos directly visible in physical space, with a compact phase-plot inset (x vs. ẋ) for readers who also want the abstract view.

⚙ Under the hood

2D Duffing-oscillator lab with RK4-integrated equations of motion, a real-space double-well potential view, a phase-plot inset, and a twin trajectory released a hair apart to show chaotic divergence growing in real time.

duffing oscillatorchaos theorydouble well potentialsensitivity to initial conditionsphase spacerunge-kutta

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install