The Duffing equation is an ordinary differential equation describing a nonlinear damped driven oscillator.

If the parameter μ were zero, this would be a damped driven linear oscillator. It’s the nonlinear *x*³ term that makes things nonlinear and interesting.

Using an analog computer in 1961, Youshisuke Ueda discovered that this system was chaotic. It was the first discovery of chaos in a second-order non-autonomous periodic system. Lorenz discovered his famous Lorenz attractor around the same time, though his system is third order.

Ueda’s system has been called “Ueda’s strange attractor” or the “Japanese attractor.”

In the linear case, when μ = 0, the phase portrait simply spirals outward from the origin towards its steady state.

But things get much more interesting when μ = 1. Here’s Mathematica code to numerically solve the differential equation and plot the phase portrait.

duff[t_] = NDSolve[{x''[t] + 0.05 x'[t] + x[t] + x[t]^3 == 7.5 Cos[t], x[0] == 0, x'[0] == 1}, x[t], {t, 0, 100}][[1, 1, 2]] ParametricPlot[{duff[t], duff'[t]}, {t, 0, 50}, AspectRatio -> 1]

Here’s the same system integrated out further, to *t* = 200 rather 50.

Source: Wanda Szemplińska-Stupnicka. Chaos, Bifurcations and Fractals Around Us.