Driven Damped Duffing Oscillator

What you are seeing: a ball in a double-well potential being pushed back and forth at a fixed frequency. Without the push the ball settles into one of the two wells; with a small push it rattles around in one well; with a strong enough push it occasionally jumps over the barrier between the wells. The Duffing equation $\ddot x + \delta\dot x - x + x^3 = \gamma\cos(\omega t)$ is the standard textbook example of a chaotic driven oscillator.

The left panel shows the phase portrait $(x, \dot x)$. Dots: $x$ vs $\dot x$ sampled once per drive period $T = 2\pi/\omega$ ("strobe", or Poincare section). A clean periodic orbit lays down a single dot; period-2 lays two; chaos sprays a fractal cloud. The right panel is the bifurcation diagram: for each value of $\gamma$ on the horizontal axis, we plot the strobed $x$ values vertically. You can see the period-doubling cascade as $\gamma$ crosses about $0.4$.