Duffing equation

The Duffing equation (or Duffing oscillator), named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by

where the (unknown) function ${\displaystyle x=x(t)}$ is the displacement at time ${\displaystyle t,}$ ${\displaystyle {\dot {x}}}$ is the first derivative of ${\displaystyle x}$ with respect to time, i.e. velocity, and ${\displaystyle {\ddot {x}}}$ is the second time-derivative of ${\displaystyle x,}$ i.e. acceleration. The numbers ${\displaystyle \delta ,}$ ${\displaystyle \alpha ,}$ ${\displaystyle \beta ,}$ ${\displaystyle \gamma }$ and ${\displaystyle \omega }$ are given constants.

...
Wikipedia